It will still depend on the supply and demand. So, the price fluctuations will still be normal because Bitcoin is a volatile and speculative asset. There could be some changes when all of the bitcoin has been mined. But, I am not thinking of that because I am sure that I won't be able to witness that when that happens. And as for the miners, their main job is to keep on confirming transactions at that time.
Indeed, if we think about the future where BTC has finished mining, that's certainly a very distant time, and of course, I personally don't necessarily think I'll still be around when BTC is finished mining, as it's rare for people to live to be hundreds of years old these days. Hopefully, we can live to that point when BTC is finished mining, and may God grant us physical and spiritual health until then.
I also think the miners will confirm that at that time, so yes, BTC will certainly continue to exist, and the miners will certainly continue working.
We won't be around that for sure or if we still are, we'd be part of the records of our localities where we might be the oldest villagers ever living. I know that it's impossibility to be still at that time around 2140 but yes, who knows - God knows if he's going to give us that life of more than 100 years. But because of the mortality rate and the life expectancy of people, currently in my country it's just around 72 years old and more than that is considered as a bonus already.