How many times can a key be used for HMAC-SHA256 before it needs to be rotated? If I wanted to use hmac for proving that a value hadn't been tampered with, how many times can I use a key before it needs to be rotated?
There is no known inherent weakness in HMAC-SHA256 that requires key rotation for a suitably secret key. 

However, other aspects of the system in which you are using HMAC-SHA256 may make key rotation necessary. For instance if you don't want the same message to have the same hmac over time for some reason a key rotation would achieve that (but so would a salt). Or if you know your key comes from an unusually low entropy pool making it easier to brute force than one would expect.