How secure would HMAC-SHA3 be?
The Keccak submission says:


  From the security claim in [12], a PRF constructed using HMAC shall resist a distinguishing attack that requires much fewer than $2^{c/2}$ queries and significantly less computation than a pre-image attack.


Here, $c$ denotes the capacity of the sponge, i.e. the effective size of the internal state in bits.

Since HMAC is a deterministic iterated MAC (in particular, it does not use a nonce), it is always vulnerable to a generic birthday-based existential forgery attack requiring on the order of $2^{c/2}$ MAC queries (Preneel &amp; Oorschot, 1999).

Thus, the claimed security level of HMAC-SHA3 is the same as the overall maximum attainable security level for HMAC, or any other deterministic iterated MAC construction, with the same effective internal state size.