Are HMACs based on hashes with larger bit-lengths also more secure?
HMAC-SHA-256 is sufficient for up to 256 bit security. Confer e.g. NIST SP 800-107. This recommendation is based on the premise that collision attacks are infeasible against common uses of HMAC, and that you consequently only have to worry about primary pre-image attacks that attempt to recover the secret key (and use this for forging subsequent messages). 

SHA-256 has 256 bit security against primary pre-image attacks. This is sufficient for all practical purposes.