The Chess.com app uses a haptic feedback feature that vibrate when a move is made, and when the time is low in your game.

But if you don't like these, you can also turn one or both of them off! ​

For the vibration on every move, go to your settings page by tapping 'more' on the bottom of the home screen in the app, then tapping 'Settings', and from your settings menu select 'Board', and then you can enable or disable haptic feedback:

Or, if you want to disable the low time warning vibration, you will find the setting in 'Play', and then 'Live Chess: