Pets obviously don't do anything on Leviathan
Ignis is completely pet-friendly
Razorscale, keep your pet on passive and by your side when she's in the air or she will most likely stand in a fire or get whirlwinded, but fine when she's grounded
XT-002 is completely pet-friendly (though if you have your pet on passive, he'll keep running back to your side every time XT goes into or out of a heart phase, slightly annoying)
Iron Council requires a little bit of pet micromanagement so it doesn't stand in a Rune of Death, Overload, or Lightning Tendrils
Pets still act a bit buggy on Kologarn, not wanting to attack him and such, but he's fairly pet-friendly otherwise (just don't let pet stand by the adds that spawn when an arm dies, or get zapped by lasers)
Auriaya, Hodir, Freya are all completely pet-friendly except that they'll occasionally get pounced on by Auriaya's adds
Thorim has WWing adds and sometimes gets pets with Chain Lightning, I don't think mine usually survives that one
It's not even worth bringing your pet out for Mimiron, even on passive
Vezax is completely pet-friendly
Not sure about Yoggy, haven't made it past phase 2 yet