There's at least two unique cutscenes you get by long resting before the grove, one after Nettie and/or Zorru, not to mention early camp interactions you miss out on by not long resting early and often in general since act 1 has most of the camp events in the game.

Personally I usually pick up Shadowheart, Astarion and Gale, long rest (well, partial rest since I have no need to recover spell slots etc. at this point usually), get Lae'zel, get Withers, long rest, go to the Grove, do stuff there, then long rest (if I'm playing Durge I go see Alfira after this long rest just as a precaution). After that you can be a little more lax about stuff, I think, but long resting is still important in act 1. By act 2 it becomes less important for companion quest progression, and in act 3 there's only a few camp events.

Basically a good rule is that you should long rest after doing main quest stuff and companion stuff (recruiting Karlach has an immediate unique camp event for example).

You should have enough camp supplies by the end of act 1 to more or less carry you through the entire game, but if you don't want to use them and you feel like long resting is pointless to recover hp/spell slots etc because you're full on those, just partial rest. You get the camp events but use none of the supplies.

Also regarding this:
Originally Posted by alceryes
Are there any timers that long rests works against? (things I can miss if I become a narcoleptic protagonist?)
There are a few times when long resting affects things, like you go to sleep right next to a fire where people are screaming for help or in the Grymforge before dealing with the situation there. I think the game even warns you about that one. Most of the time you can do whatever you want, though.