To me, a dungeon crawl is a mechanics-first strategy-focused RPG that differs from a tactical board game both in its typical level of complexity and time commitment, but more fundamentally in its rewarding of player creativity. 'Plot' and 'Story' are completely irrelevant in a dungeon crawl and, while players do have characters and character interactions, these are vehicles for the strategic looting of the dungeon (and possibly the defeating of the BBEG, or other MacGuffin). My point is that, in a dungeon crawl, the entire narrative framework is a MacGuffin to drive the player characters' into challenging situations. The game itself is about the challenging situations and how the players- and only to a lesser extent the characters- overcome or fail to overcome them. The cost of overcoming obstacles is relevant, but the emotional impact of overcoming them is not. This is very much contrary to the fundamental concepts of most modern RPGs, which tend to be about characters, story, philosophy, or something else where the PCs don't really 'win' or 'lose'. In tactical RPGs (aka dungeon crawls) players really do win and lose (or, on a single-session level, lose or not-lose-yet, which feels more like winning the closer to losing you came); they 'win' when they 'beat' the campaign (which, if properly designed, has realistic odds of killing their characters, will kill their characters if they play stupidly, and will or has a chance to reward parties that overcome the campaign with exceptional success) and they 'lose' if their character dies (unless death was a deliberate decision on the part of the player typically to ensure the otherwise unlikely completion of some even more important goal, like the campaign being beaten or the party's survival. TPKs are always a loss).
If this kind of complex, mechanically driven, conflict-focused co-operative role-playing game is what you mean by 'dungeon crawl' as well, no, FATE is not a good choice of system and is, in fact, one of the worst systems you could choose.
FATE is both Soft-Rules and Rules-Light
Tactical RPGs want to be hard-rules rules-heavy systems.
In the first case this is to make the game possible by establishing rules by which the GM is always bound, so the players can exploit mechanical weaknesses in the GM's plan without worrying about the GM ruling that 'things just don't work that way' or some variant. Hard rules systems encourage GM trust and respect because, if the players know the rules of the game (which they must to a fair degree to play a Tactical RPG well, or at least play with other players who do), they can see that the GM follows the rules and use unusual system behavior to extrapolate that something weird is going on rather than the GM just deciding to ignore the rules. For example imagine a system that requires an average NPC to get a roll of 10 on a d20 when drinking alcoholic beverages to avoid immediately passing out from drunkenness per drink, regardless of how many drinks are taken. If the PCs observe someone drink a 6 pack of beer over the course of a day, in a hard rules game it would be appropriate of them to assume that the character is not an average NPC wheras in a soft rules game it would be appropriate to assume the GM decided the instant-knockout-beer rule is stupid. Being able to make these kinds of inferences reliably, consistently, and independent of the GM is important in a Tactical RPG.
A rules-heavy system is preferable because, in a Tactical RPG, more rules = more character options (though not all of these options are necessarily good). FATE has a uniform conflict resolution system that applies to all situations. This is good for some kinds of player groups, though not required for any particular genre of RPG, but awful for a Tactical RPG: all FATE characters are the same character, from a tactical standpoint! (ok, technically you could vary whether your skills/aspects are more focused or more broad, but the tactically optimal choice is more focused and this is clearly stated in the rulebook, at least for 2nd edition). Hackmaster, as an example of a system designed for Tactical RPGs (well, technically, to parody a Certain Game and its culture) , adds extremely niche special-case rules deliberately, in order to promote more varied, specialized characters. Most systems don't go quite that far, but good Tactical RPG systems are almost always rules-heavy.
Early FATE is a character-focused game. Later FATE is more narrative focused. All FATE systems try to get as far away from being mechanics-focused and detail-focused as possible. The systems, especially 2.0 in my opinion, are excellent, groundbreaking works of art that revolutionized how people play narrative RPGs and how designers make narrative systems. That doesn't make them good for everything. FATE is designed specifically to discourage dungeon-crawling player behavior and dungeon-crawling gaming approaches, and it does a good job. When you use FATE for a real dungeon crawl, you are taking a finely crafted precision glass-blowing instrument and using it to smash down a door. There are other tools for door smashing. Use them.
You can certainly play a narrative-focused, rules-light game with easy conflict resolution in a medieval fantasy setting that involves the exploration of underground monster-filled ruins in FATE. You can do that quite well, in fact. But it would be easier and better to run a Tactical RPG, a Dungeon Crawl, in no system at all than to do it in FATE. And that's saying something.
As for personal experience, I have tried this, long ago, when I was young and foolish. It went badly but turned into a fun character-focused game.
I have seen other people attempt this, and it usually doesn't even go that well (I was lucky and had good players when I made my mistake).