What happens when a miner does not claim all fees/generated coins?
The block is accepted, and the coins are lost. Poor miner.

Here is a link to the part of the 0.6.3 source which checks this: https://github.com/bitcoin/bitcoin/blob/v0.6.3/src/main.cpp#L1362

Also, this has effectively happened before; block #124724 claimed one satoshi less it than could have.