Does any use Dish with Apple TV?

You would connect the Apple TV box to one HDMI port and the other tuner box on another port. You would then use your TV's (or a universal remote) to switch between inputs on the TV.

The other recommendation above was referring to having your service from Dish transferred to Sling TV. Sling has an app available on Apple TV so then you would be able to access it directly.

No, you wouldn't need a splitter, that is used to have the signal of one device going to more than one TV. If you were limited on the amount of HDMI ports your TV has then you would be using an HDMI switch.

Which Dish is that? The U.S. company Dish Network of Colorado?

Perhaps see if you can have your service transferred to Sling TV, as that is from the same company. Sling TV has an app for Apple TV.

Apple TV is just another HDMI device. You can switch between Apple TV and other tuner box (connected to a different HDMI port) with the push of a button, if you want to use these side-by-side.

I would recommend using the AppleTV remote or buy yourself a good quality Universal Remote. That way you can put away all of your remotes and use just one to control all of your devices. Personally I use a Harmony, they are simple to program (they have an online app that is free) and yes, it will cost some money but I think you will appreciate having it, that way you don't have your TV Remote, your Dish Remote, an AppleTV remote etc...