Why would it ever stop mining for coins?
In the future fees will be the deciding factor in whether mining is worth it. In the extreme (120 years) right after a block is solved there will be no reason to mine because all fees will have just been awarded (or maybe not I guess) and miners will wait until more fees are queued up before they devote their power.
A neat side effect is that the variance in block time will go down. And the average time to next block will be lower in busy times and higher in off-peak hours.