Is an Initialization Vector necessary if using different keys for each encrypted data set?
It depends on the mode. For ECB mode, no. In fact ECB mode does not take an IV. However, as soon as you encrypt data that is larger than the block size, you loose semantic security. That is, anything over the cipher's block size will leak information. See the picture of Tux at Block Cipher Modes of Operation.
Other modes, such as CBC, OFC, FBC, CTR and the others require an IV. The IV requirements differ among the modes, though. Some allow a unique IV, others require random IV, and some forbid reuse of an IV within a key.
As for your warning: just use a random IV with each encryption and you will avoid most of the pitfalls. If you re-encrypt a decrypted message, then use a fresh IV.
Is there an advantage to still using an IV or a disadvantage to not using an IV here?
Two messages encrypted under the same user will produce the same cipher text if the messages are the same. You've leaked information, and lost semantic security.