Do you have to have wi-fi to use Apple TV with your Mac or iPhone?

Not to use it with a Mac if that has an ethernet connection as well as AppleTV, but normally essential for iPhone/iPad if you want to control it with Remote app or use Airplay as iPhone/iPad do not have ethernet and could not connect to it locally - unless you have modern iPhone/AppleTV capable of peer to peer Airplay.

You could in theory use internet sharing to create a local wifi on the iPhone itself but it may not work well and some say they had heavy carrier charges for internet usage.

Use AirPlay to wirelessly stream content from your iPhone, iPad, or iPod touch - Apple Support