I don't think so, but there are some qeusts, that take a few days for the result/reward to show up, and there is the growing that takes 3 days too. Also another vendor shows up during the weekend specifically, so day tracking does play a role (and I can only assume there will be more of that kind of stuff when we get to a full release whenever that is)
Gummiel’s pretty much listed all the day related things. There aren’t any time limits or things you’ll miss if you’re too slow, although if you stay awake too long (3 AM), you’ll pass out and wake up back home.
There’s also 2 NPCs who won’t be available until day 10 (I think) who live on the west side of Port Beren - although I’ll be making things like that a bit clearer soon.