Nothing, they just lose access to them and they are out of circulation.
It can how ever increase the value of other coins.
For example, let's say I'm doing a bake sale and I have 100 pies. You come over and want to buy one, and I say $8 and you give me the money and the next person comes over.
What if I had one pie? The first person comes up and I say $8 and the one behind him offers $9 and the person behind him say, "No, I'll give you $10!". That's scarcity, by removing tokens out of the supply you potentially increase demand due to scarcity.
Although this is not always the case. Take Hive, there is almost 375M Hive in the supply, if I burned one, it won't do anything to the price. No one cares. If I burned 300M, there would certain be some reaction.