I am starting to forget the little details... but I thought it was like this:
If there's no specific note for the Fireplace or Cupboard, then those special rules apply to the first hero who searches that room for treasure only.
if I recall correctly, when we played First Light at GenCon, Zargon Doug let me heal myself with the Hearth one time. On the next turn I tried to do it again but it he said it was one use only. The healing might have been each hero who searches rather than just the first (I forget if he told me I could heal or if I attempted because someone else did it first). I don't recall if I could have also searched for a Treasure card on the same room, or if that used up my search in that particular room. Talk on the official AH discord is that the special search with the furniture is a substitute for your search in that room, and any other searches (by other heroes) will be regular treasure card draws as normal (don't think Doug has chimed in to confirm yet, but maybe he will once he gets a spare moment in these next few days/weeks).
This got me thinking about the other types of rules that might appear in the First Light Rulebook. It didn't seem like the book was that huge (I didn't handle it myself, but just visually I mean). Yet, all these tweaks and rewrites on the cards they did show makes me wonder what other things might be tucked in there...
Probably anything mentioned on the revised (front & back) "Reference Cards" would be in the booklet, probably the same with any notes presented on the Zargon GM screen.
It wouldn't be necessary to make the rulebook multiple times larger than the original by including every single rule that appears in every single expansion of course, only the ones that would be used in FL itself, though we've seen several new rules that are sort of implied they can be used anywhere, and might enhance the experience for players, whether they are "optional" or not. I am talking here about general rules, not card tweaks (like saying the Hand Axe is not usable by the Wizard or the Helmet is considered "metal armor" as we've already seen on the cards in FL as well... or the fact that we have a new monster type in the "Dragon" which is used multiple times throughout the 10 FL quests). First Light of course is meant to be compatible with all the official expansions, so it's not necessary to retread everything, but only those rules they may wish players to feel free to start using from the beginning.
Possible Candidates for FL Rulebook inclusions:

Arena Mode/Tournament rules: It's already know that these rules are in fact used in FL, so despite the lack of "hourglass" (activation tiles) present in the GenCon demo, that these rules introduced in ATOH are present in the questbook and/or rulebook. The Arena with its large 7x9 room and two 4x5 side rooms is a permanent fixture of the "Caverns" board on the reverse side of the FL gameboard. [Edit: the arena room is slightly smaller and the full tournament rules aren't used compared to ATOH remake]

Monster Substitution: "What happens if I run out of Monsters?" is a questioned answered in the Zargon sections of the original rulebook already as substitute a monster of the same color. FH/MOTM revised this to say a monster of similar strength (granting that these quest packs had most monsters being of the same pastel shade of blue hence it might throw things off to place a Giant Wolf or the Frozen Horror when only an Ice Gremlin or Elven Archer was originally called for!). First Light however does not use monochrome plastic miniatures for its monsters (except the new dragon, which is a unique silvery dark gray color) but full color cardboard standees.

Entrance/Exit Doors: One of those very basic additions that often gets forgotten... KK & ROTWL introduced that many quests do not begin and end on the Spiral Staircase tile (or just automatically end when the mission is completed such as killing a particular boss monster) but instead begins on the edge of the board with an "Iron" (or Steel) entrance door and often concludes with a Wooden exit door on another board edge. KK/ROTWL started the trend of including extra such special doors for this usage. Some expansions however called for this rule but didn't include the special doors (Zargon can still just use any closed door to represent it, and then re-use the door later if he runs out). ATOH also came with those large arena doors which could serve the same function (but didn't include the iron/wooden door variant miniatures).
Fun facts: classic era HQ simply used cardboard inserts to represent the special doors, and these were just inserted into any standard door base. The Brazilian edition of HeroQuest from 1993 used a slightly smaller version of the gray plastic door base for all standees (which in that version included the heroes and all monsters). While AH's remake of HQ uses sculpted doors along with other sculpted furniture, those gray plastic door bases saw a return in MOTM and ROTDM as the mirror bases! And now First Light uses transparent and round plastic bases for doors and monster standees. 
Zargon's Imagination/Experience: MOTM/FH mentioned this "rule" which may be the first official public sanction in print of the designers' intention for Zargon to not only run the game according to the regular rules (and things like remixing quests when they are replayed, introducing lost Artifacts as special treasures early in the next adventure) but he is free to improvise and decide on things when the situation calls for it (like an ambiguity or gap in the rules or story). Zargon is the GM... the conflict resolver and his word is law at the table! This is a great one to include, regardless, especially for players new to this genre of game and not only because not every rule can be clarified without making the rulebook many times larger than it is!

Treasure - seems like a basic one, but pretty much every expansion and digital quest advises that large gold rewards should be divided among the surviving heroes to prevent conflicts. In theory I guess this applies to any gold reward, not just those presented between quests (which are almost always divided among the heroes, though sometimes they are granted to a hero who achieves a specific goal).

Selling Equipment - First mentioned in FH/MOTM but subsequent expansions as well.. between quests heroes can sell back their unwanted Equipment back to "the Armory" for half the listed gold value. Since AH's remake puts things like Potions on Equipment cards, those in theory would follow the same rules (but nothing is mentioned about Alchemy cards.. except the ones that list a specific gold value on them must theoretically be sold back... consult ROTDM's rules for more on Alchemy cards... I didn't see any Alchemy cards or Re-agent Treasure cards in the FL demo shown at GenCon so I'm assuming these rules won't be part of it).

Passing Items between Heroes. Featured in New Beginnings showing it was intended to be used from the start, not just in MOTM/FH where it first appeared.
Selling back Equipment. Same as above.

Shock - Mind Points: The "shock" rule seems to have been made universal as what happens to a hero with zero mind points in all cases except two... ROTWL where it means the hero is unconscious and out for the duration of the quest... and KK where it means death unless someone has the Elixir of Life to revive them. Even ATOH which originally ahd the unconscious rule in the original version adopted the Shock rule in the remnake. Into the Northlands clarified that anything that restores even one Mind Point brings a hero out of "shock." The Game System had no need for such clarifications because there was nothing to take away mind points (the only other expansion where you didn't lose Mind Points when playing "as written" was MOTM).

Mercenaries... unlikely to be mentioned since Mercenaries are only used in certain quests, then again in theory they could mention it especially if it's tied to...

Animal Companions: First mentioned in Into the Northlands (and then subsequent quest packs ATOH and JoD), when there are less than four heroes in a quest, each hero can hire one animal companion for free to accompany them. The character is more limited than a human mercenary but pretty strong, and can be represented by a single tile (such as the "mystery tile" of the Mentor Eye/Zargon Skull symbol included in the GS and First Light... KK and ROTWL also included unique "mystery tiles").

Unthreatened Movement: First mentioned in Into the Northlands as an optional rule from AH's own playtesting... with no monsters on the board, Heroes can choose to just move up to 4 squares for each red movement die they would normally roll, instead of actually rolling dice. Thus a 1d6 hero could move 4 automatically, and a 2d6 hero would be able to move 8 (how spells affect these is up to Zargon).

Multi-Attacks: FH introduced the ability of Polar Warbear monsters to attack either two heroes or the same hero twice. In the case of a double attack, Into the Northlands clarified the intention was that if two attacks are launched against the same target, then the victim not only gets to roll one defensive roll but has to choose WHICH of the two attacks it will defend against... the other remains unblockable. Rogue Heir of Elethorn (the 2nd hero collection) clarified this same sort of rule should apply to hero multi-attacks against monsters as well. This wouldn't just apply then to the Rogue's special skills, but to any hero using Heroic Brew or Orc's Bane (the FL card clarifies Orc's Bane can only be used against a single orc, so it's always a multi-attack under this version's rules).

Turn Order Option: Jungles of Delthrak allowed the heroes to observe, in a new "round" of turns the changing of their character turn order. This is a nice one especially if the heroes realize they goofed up at the start (such as by putting the Wizard last!).

Incapacitated Option: Introduced in Jungles of Delthrak, "death saves" can be handled the usual way (zero body points hero is dead UNLESS he has a healing potion/item or unused healing spell - assuming no action was taken on his turn already)... or in one of two other ways... "Heroic:" a zero BP hero is only "incapacitated" and unable to do anything until another hero comes along adjacent to save him with a potion or healing spell (I think regardless of whether they have used an action on their turn, they can still use the healing spell to revive the downed character) - but a skull time is placed on the square where the hero fell indicating they have only one "round" of turns to save them or they really are dead... and the third option called "Story" in which its the same as before but there's no skull tile placed - meaning the heroes only lose if they're all incapacitated and thus no one is left to save them, plus if there are no monsters on the board each incapacitated hero regains 1 lost BP meaning they are back to normal.
Will the Wizard be advised to "save his gold"? This advice seems wrong unless they plan to release something like the Wizard Quest Pack where the Wizard can indeed purchase "powerful magical items" that only he can use (the 1990 game system implied this but by 1994 plans were scrapped to release this and the "lost" Dwarf quest pack, sadly). Instead the Wizard ought to give his gold away... if he's not going to spend it on Potions and Mercenaries of course since otherwise all he can buy is the staff, toolkit, and daggers, though the Wizard's Staff is superior to a regular one.
The "Orc with the Large Sword" was called for in the GS to represent various Orc type named or boss characters. We've seen that while this was a bit clumsy in the remake (since there were two Orcs with large swords in the box rather than the one of the original, it usually didn't present a problem since you would typically have enough Orcs to use other sculpts to represent "normal" Orcs next to the special one). The "orc with the large sword" is represented on the cardboard monster standees because there is a standee equivalent of each plastic monster that would be found in the 2021/2022 GS box but once again there are two of them. This is a questbook issue rather than a rulebook one, and really doesn't seem to matter as a look through the adventures seems to show that there are no special characters so designated in this campaign.
Any others I missed?
I'm doubting we'll see any clarification of what squares precisely the Crossbow can hit (though they could easily mention that in the Line of Sight pages) and I don't recall seeing any specifically clarified wording on the Crossbow card (photos of those weren't allowed at GenCon).