As written, the level up move in Dungeon World says that one must "spend some time (hours or a day)" in order to level up. Why is this the case? What mechanical or fictional benefit does this provide? I am debating removing this clause from the move because I basically don't plan on giving my characters any time to rest for the next 5 sessions. Would this have an adverse effect on my game?
The intent is that players level during a moment of rest and reflection
This mirrors both the old school D&D feel that the setting is based on and many other fictional sources where the heroes catch a respite and train or think about what they've experienced and apply new ideas. Mechanically this does prevent you from leveling if you are actively running around a dungeon skirmishing and looting despite the fact that you may be at or over the required XP.
House rules are fine though
That said I don't feel like your houserule will in anyway break the game. If you are working on a dungeon or other scenario where the players will be active and engaged against increasingly difficult threats then the need to level up as they go makes sense. Likewise I've been a player in a game where the GM houseruled in the opposite direction that players could only levelup if they did the Carouse move.
Because Dungeon World is intended to simulate old school D&D, and to an extent the fantasy fiction it represents, it wants some downtime to advance. In those sources you don't "ding" like in a video game once you get enough XP, you have to train and such. If you are not interested in simulating that kind of fictional world, you can certainly change it and simulate something more video gamey with practically zero game rules impact. It does have world feel impact, which is what DW is primarily concerned with.
However I am concerned that you have another problem that may be interacting with this - "I don't plan on... for 5 sessions." DW's intent is to be player agenda driven - play to find out what happens. If they need a rest to level, they'll find a way to do it - "planning out your plot" to the degree of what specifically they'll be doing in any given session is likely to break a number of DW mechanics because they are predicated on a different set core assumptions (see http://www.dungeonworldsrd.com/gamemastering/#How_to_GM) for the details). Though you may prep, DW says that you should not be planning everything - and that's a pretty long time horizon for detailed planning even for a plain ol' trad D&D game, let alone Dungeon World.
This is how you play to find out what happens. You’re sharing in the fun of finding out how the characters react to and change the world you’re portraying. You’re all participants in a great adventure that’s unfolding. So really, don’t plan too hard. The rules of the game will fight you. It’s fun to see how things unfold, trust us.
The only reason it's there is to enforce a bit of longer pacing on adventures. It makes the players want to stop and rest. (The alternative is that they may accidentally never have the PCs sleep, and then you've got a GMing problem where you have to show them the side effects (via GM moves like Unwelcome Truth) of not sleeping.)
However, it's really unnecessary to reduce the amount of time — presumably your group's party will eat and sleep, and there will be hours in there which will trigger the move, without having to alter it.
The only reason you'd need to alter the move to make it work in a no-rest game is if you literally allowed them no opportunities to rest.* In a game where (for example) they're in a literal whole-world dungeon and literally every time they turn around a new thing is trying to kill them, and the only rest they ever get is a minute to breathe and they only ever ate on the run because to do otherwise meant likely death, then it might make sense to alter the move. (That sounds like a tense and interesting — but extremely unusual — game of Dungeon World!)
As for the side effects, you'd only have two: players would no longer be motivated by the move to find opportunities for rest, and would charge forward into adventure more; and you'd have them level-up in the middle of whatever's happening (combat, a tense conversation with an NPC, etc.). The latter is fine — it's how Apocalypse World, Dungeon World's parent game, works already and it's fine there — and doesn't interrupt much because levelling up is so quick in Dungeon World.
Short version: unless you're going to be running a very unusual Dungeon World setting, you probably don't need to change the move at all. And if you do change the move, so long as you're okay with level-ups happening mid-action, and you're okay with players being incentivised to never slow down a bit, there's no downsides.
* Which, by the way the GM moves trigger only under certain circumstances, is not actually fully under your control.
I'd like to add a tiny suggestion: leveling up during a battle can easily kill the momentum. Imagine someone rolling poorly and before you know they are out of the game choosing a new skill, which leads to:
- asking questions about skills and distracting everyone
- choosing params/skills that will help in current situation, but not necessarily in the long run
- 'cheating' by keeping levelup until the last possible moment - "well, i rolled 10 on hack and slash so i levelup to increase my damage dice size by one"
And all this can affect the game negatively. I'd suggest at least a minute of quiet time for level up move to happen. OR you can ask your players to have 1-2 levels figured out before the game so it can be done seamlessly.