What are some disadvantages of homomorphic encryption schemes? I'm doing some self-teaching / research for my own benefit in homomorphic cryptography.

I've studied both additive and multiplicative schemes (Paillier and RSA respectively), but all I can seem to find are the benefits of the schemes.

Are there any disadvantages to these schemes being used for homomorphic operations (minus their obvious limitations in not being fully homomorphic)? If so, where should I be looking, and which key-words have I not been searching for?
The multiplicatively homomorphic variant of RSA is not semantically secure. This is a major disadvantage. ElGamal is a semantically secure, multiplicativey homomorphic cipher. Paillier is a semantically secure, additively homomorphic cipher.

As described by tylo, all homomorphic ciphers are malleable by definition. Chances are, however, if you are interested in homomorphic ciphers, however, malleability is probably not a concern.

Homomorphic ciphers typically do not, in and of themselves, do not provide verifiable computing. In words, you encrypt your data, send it to the cloud and let the cloud compute on it for you. How do you know the cloud performed the correct computation? To get this sort of guarantee, other machinery is needed.

Performance is often a disadvantage. Ciphertexts in the ciphers you mention are much larger than the plaintexts, so communication requirements typically go up. The computations on these large ciphertexts are typically slower than if you just performed the computation on the plaintext itself. Because of this, in the outsourcing computation model, we typically see a requirement that encrypting inputs and decrypting outputs should be faster than performing the computation itself. In the case of multiple parties with individual inputs this seems to be less of a concern as privacy, not efficiency is the concern.

In the case of multiple, participating parties, guaranteeing fairness (which means everyone who is suppose to get an output, gets it) is often difficult and requires extra machinery (e.g., threshold decryption) and more assumptions (threshold of honest parties, etc). Another concern in the multiple, participating parties model is how the parties privately input their values for the computation. If one party knows the private key, they can decrypt the inputs and violate another party's privacy. So in this case, threshold decryption is often used. This in itself is a disadvantage as generating threshold keys is a non-trivial task.