The transaction fees are claimed in the coinbase, so a party refusing to accept a miner's proceeds (by boycotting a corresponding protocol change) means also refusing both the block reward subsidy as well as any transaction fees.
I gather you are not referring to Coinbase
https://coinbase.com/ specifically but rather
the database of Bitcoins, right? Does the coin database (is this the block chain?) grow without limit or is there some sort of truncation eventually?
"... a miner's proceeds ..." -- If I understand then you mean the miners will continue to mine even after the last of the 21,000,000 Bitcoins are mined. They will "mine" not for new coins anymore but instead just for the, um, confirmation/validation/encryption of the new blocks of transactions. Neat. So, as long as new transactions continue to flow the miners can still make a living.
Hmm, two miners (unintentionally or deliberately) creating divergent block chains force entities to make a choice; the majority win. Entities should always desire to be on the winning side otherwise they risk becoming the victims of double spends.
In the earlier moments of a hard fork (who's job is it to be watchful and broadcast the alert?), it might not be clear at all which side will become the winning side. Entities (at least those in the know) would be wise to wait (for how long? what if a paid service is life critical?) for the situation to settle down. Hmm. It feels like waiting is not acting with authority.
In theory there might not be a central authority per se but in practice continuity of support can potentially provide quicker and higher quality results reducing the risks to the Bitcoin community.