So, if a miner will send a transaction, and will put some coins as a fee, then the same miner can simply take less amount in the coinbase transaction, then those fees will be burned. That's why miner-based burning is not limited only to the new coins.
OK, this
is an interesting approach. Not such a common practice, but it is possible; actually I think that there were some coins burned (kind of) like this - most probably due to miners' mistakes while experimenting.