i'm not sure but it will happen!
Run out
(of a supply of something) be used up.
If you mean would bitcoin supply dry up eventually, yes it would, probably around 2140. But it cannot be exactly predicted when the last bitcoin would be mined because 100 years from now technology would have changed a lot and how it is going to affect mining, no one knows.
As it is very difficult to predict how mining power will evolve into the future - i.e. whether technological progress will continue to make hardware faster or whether mining will hit a a technological wall; or whether or not faster methods of SHA2 calculation will be discovered - putting an exact date or even year on this event is difficult.
https://en.bitcoin.it/wiki/Controlled_supplyEven after the last bitcoin has been mined, it is not going to change anything. That is why halving is there, started from 50 blocks and gradually decreasing so that miners could easily transition from incentives from block reward to incentives from bitcoin transaction fees.