Is there a key that makes AES the identity function?
No, probably not. That would mean that exactly the right permutation would be chosen out of the almost infinite set of permutations that are possible.
Is there a key that makes AES the identity function for certain inputs?
That's more likely, but it won't be easy to find it.
Is it known whether such keys might exist?
Not to my knowledge. I cannot answer for the rest of earths population though.
Are such keys known for other ciphers?
What you are probably looking for is the notion of weak keys.
The Enigma already had specific methods of avoiding to encrypt the plaintext to the same ciphertext. A modern block cipher would be considered broken if it had a key that would result in the identity function; a modern block cipher isn't supposed to have weak keys. The output would be clearly distinguishable from random, even if just for that key.
Now the chance that a single plaintext would result in the same ciphertext is of course related to the size that is being encrypted. For block ciphers such as AES that would be the block size. It's easy to see that the chance that a particular plaintext encrypts to itself with a certainty of $1/{2^{n}}$ given a specific key.
There are however also $2^{n}$ possible plaintext and of course $2^{k}$ keys (where $n$ is the block size and $k$ the key size). This means that it is extremely likely that there is a key / plaintext combination that would result in the permutation having one relation between plaintext and ciphertext where both are the same.
The trouble is that block ciphers are generally constructed in such a way that this combination would be extremely hard to find. Of course, with DES having a certain kind of weak keys, double DES would be a prime candidate for creating a block cipher acting like a identity function. You just double encrypt with a known weak key and presto: identity function.
The AES competition however required the candidates to have fully random keys and - of course - no weak keys [citation needed]. Combine that with a relatively high block size / key size and it may take quite a bit of a search to even find one key with a single "identity relationship".