Mimics are obviously most renowned for taking the forms of chests. However, after hearing that they can polymorph into other furniture, I wonder do doors become a possibility?
If yes, what are the limitations to a mimic's shape-changing abilities?
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Mimic, MM p.220
... these cunning creatures most often take the form of doors and chests ...
As for the limits of a Mimic's abilities? Up to the DM, as dictated by the needs of the encounter, story, and whimsy. Some guidelines are in the Monster Manual.
The rules don't exclude this, and the description suggest that they do this.
5e Monster Manual (p. 220) says the following in the mimic's rules:
The mimic can use its action to polymorph into an object or back into its true, amorphous form. Its statistics are the same in each form. Any equipment it is wearing or carrying isn 't transformed. It reverts to its true form if it dies.
Furthermore, its other two rules (Adhesive and False Appearance help it disguise itself, the latter to make it look like a door and the former so that it can squeeze itself inside of the door frame. Furthermore, the mimic's description (which I must note is just that, and not a ruling) says this:
the form of inanimate objects to lure creatures to their doom. In dungeons, these cunning creatures most often take the form of doors and chests, having learned that such forms attract a steady stream of prey. [...] Mimics can alter their outward texture to resemble wood, stone, and other basic materials, and they have evolved to assume the appearance of objects that other creatures are likely to come into contact with. A mimic in its altered form is nearly unrecognizable until potential prey blunders into its reach, whereupon the monster sprouts pseudopods and attacks.
So can a mimic imitate a door? Probably. Do be wary though that if you spring this trap on your PCs there is a good chance that they assume for quite a while that every single door they face is a mimic, which can slow the game down a bit.