Answering on the topic, miners would still continue to work since they're primarily needed into the network on processing or pushing transactions and as said above they would instead increase up the fee for them to make money.
No need to increase the fee!
We are definitely going to mine out all the bitcoins one day (probably after ~150 years) and then still miners will continue their process of mining as usual. Why and how possible?
We need to consider two factors here : Price of one bitcoin and block size (or number of transaction going to happen every 10 min) after ~150 years.
Let's assume,
One bitcoin may value one billion dollar or at least 500 million dollar after 150 years and block size may remain same 1MB but segwit or anything similar enhancement may allow a block to carry some 10k number of transactions per block.
So, miners will get some 5X more fees than what they earn today and its worth may value 50,000X than today.
So, in rough calculations mining will be profitable at 250,000 more times than today.
So, do you still think about miners to increase the tx fees or about stopping due to non-profitable after all 21 million bitcoins to be mined out?