Toilet flush button is corroded and gets stuck
I happen to like pure silicone grease for articulating plumbing parts. I keep a jar of food grade silicone grease in the shop. Its usually sold in scuba gear shops, but I see it gaining popularity for other applications.

I'd clean the toilet flush button parts with something like CLR, lube with silicone grease and reassemble. Besides lubricating, the grease should help keep air and water off of the brass to slow down the corrosion. But it's probably time to start looking for some replacement parts.