Almost definitely nothing.
The rest variant you have described, where a short rest is 8 hours and a long rest is 1 week, is exactly as described in in the Dungeon Master's Guide on page 267 in a section on Rest Variants.
The designers suggest this option for "gritty realism" and do not provide any warnings about what this might break or imbalance, and their silence suggests that nothing unexpected* should happen if you implement this rest variant. In other words, it should be safe to utilize this variant without anticipating the need to fix anything else and without endangering the integrity of your campaign as long as you plan for adequate opportunities to rest and avoid overloading the players with a combat pacing that presumes the standard rest schedule.
(* By "nothing unexpected," I mean that obviously your characters will be more hard-pressed with their resources, but that is an expected consequence of this variant. You shouldn't see any game-breaking side effects, however.)
I'm currently running a West Marches style game. I realised before I even began that, if I were to use the standard resting times, I would either have to run an excessive number of random encounters, or have each and every one steam-rolled by a fully-rested party.
My solution was quite simple. I changed short rests to 8 hours (and renamed them to simply 'rests'), and completely removed long rests, instead saying that, at the start of every new session, the party is fully rested. Because, in the West Marches format, every session is a new one-shot (of sorts), and every party starts and ends in town, it seems natural that the party should start each session rested (having just been staying in town).
As a result, each session contains a reasonable number of encounters (including traps, environmental hazards, etc.), such that the 'adventuring day' is in fact split over a week (or so) of in-game time. This keeps the number of random encounters each (in-game) day feeling natural (one or two), while still taxing the party's resources towards the end of the session.
It seems to be working well, although I foresee that resources could be stretched rather too thin if the session were allowed to run on longer (than about 6 hours), or an adventuring day were allowed to run across multiple sessions.
In theory, it would be possible to do this in any campaign, as long as the players were willing to accept the slight implausibility of being fully-rested at the start of every session, regardless of circumstances. The key is recognising that it is the long rest, not the short rest, which has the greatest potential to imbalance the game. Because recovering hit points in short rests is tied to hit dice, allowing your party as many short rests as they like is unlikely to radically affect balance.
To summarise: The 'adventuring day' need not be an actual in-game day. It can instead be recast as the time elapsing in one session.
Best Answer
This could lead to unbalance among the classes
This answer assumes that the party will still have a similar number of encounters per 24 hour period.
As CTWind's answer pointed out, longer days will effectively nerf any spell that has a duration of 1 hour or more and doesn't replenish on a short rest. This also nerfs almost all of the caster classes, as nearly all of them only replenish spell slots on a long rest.
Magic based classes will be weaker
A wizard, for example, is a powerful class because they are incredibly versatile and can cast so many spells between rests. But, they can't get those spell slots back without a long rest, meaning your wizards will have to be very reserved about which spells they cast and when. Classes with fewer spell slots, like the sorcerer or cleric, will get hit even harder, because they have far fewer spell slots to begin with.
Mixed classes will also suffer
"Mixed" classes that do both combat and magic, will still be impacted, but less so than the pure casters. The ranger, for example, often relies on a mix of magic and combat to be effective, though it relies far less on magic than the pure caster classes do.
Non-magic classes will probably be fine
Non-magical classes like the fighter and rogue I think will be impacted the least - especially the rogue, which often relies heavily on being able to sneak and use bonus actions, neither of which has any constraints on how often they can be used.
The non-magic classes will be stronger compared to the other classes
This will (generally) result in the non-magical classes becoming stronger relative to other classes, with that discrepancy growing the more reliant on magic the other class is.
Exceptions
There are a few exceptions though.
Low-level barbarians will probably suffer more than, say, a fighter or a rogue, since they only have a limited amount of rage between long rests. By stretching out the time between those long rests, the barbarian becomes much more vulnerable, at least until higher levels.
Paladins will also be impacted, though I'm not sure to what extent. I've never played one, but I know that Paladins are a "nova" class, meaning that they can do a genuinely stupid amount of damage all at once, but then they can't do that again until they get a long rest in. Extending the day also prolongs the downtime between paladin damage bursts, which could significantly reduce their damage output. That said, Paladins are also just really good at soaking damage and dying very very slowly, so that might not be as big an issue as I think.
Then there's the warlock. Unlike other magic-based classes, the warlock restores its spell slots after a short rest. This means that while the wizard, sorcerer, druid, and so on are carefully conserving their resources, the warlock is free to cast its spells with no regard to the longer daytime hours! As long as it can get in an hour of downtime here and there, those spells can keep on flowing! This is effectively a buff to the class, rather than the nerf that other caster classes get from this!
If you're going to do this, treat it as a playtest
I don't know to what extent this will actually impact your game. You might try this out and find that with your pace, the game isn't really impacted at all. On the other hand, your pace might lead to some really unusual balance issues as described above. I haven't done anything like this, so I don't actually know how noticeable the impact will be. So, if you decide to play this way, I suggest treating it like a playtest, and be ready to tweak the game to try to keep things balanced. Just make sure your players are alright with that beforehand!
On the other hand...
If players and DM are both fine with it, exploiting a broken mechanic can actually be a lot of fun!