YouTube on the web has supported 60fps content since October of 2014, and now it is finally coming to mobile. Google has updated the apps server-side, so no download is required. It seemed that Google was going to switch this on back in May of 2015, but that disappeared shortly after.
This update arrives both for the iOS and Android versions of the app, so you will be good no matter which platform you use. Only select videos are supported, since they need to be shot with this frame-rate in mind. Most video games are 60fps, so that is where you’ll most likely find that content.
Keep in mind that this will no doubt use more bandwidth than before, due to the amount of additional frames. If you have a low amount of data, try sticking to 720p instead of 1080p or higher. If you’re on Wifi, you won’t have to worry about your data. Here is a 60fps video to get you started: