Well, here's the relevant part of the arcane lock spell:
You touch a closed door, window, gate, chest, or other entryway, and it becomes locked for the duration.
That seems pretty clear. There's nothing there that limits the spell to only non-living materials; If it's in the list, or counts as an "entryway," it can be locked by this spell.
As for the specific examples you mentioned:
- A door is both on the list and is undeniably an entryway. Whether it's made of iron, living wood, or ever-screaming demonflesh, it will be locked by this spell.
- A tentacle or vine wrapped around something isn't an entryway of any
kind, and so won't be locked by this spell.
- A mimic is a creature, not an entryway, and so won't be locked. That being said, see the next point.
- Whether a creature's mouth or other orifice constitutes an entryway is something you'll have to ask your GM. (Try to keep a straight face when you do.) Note that even if your GM agrees that an orifice is an entryway, you may have difficulty using this spell in combat, as the spell requires you to touch the entryway in question (and not just the creature it belongs to).
As far as I can tell, the only side effects to locking a living entryway are the ones that apply to locking any entryway: It's harder for things to pass through an entryway that's locked, and anyone planning to use it will be inconvenienced and possibly annoyed.