updated 12:40 pm EDT, Wed June 18, 2014
Would likely focus exclusively on media
The Apple TV may gain some form of Continuity support when OS X Yosemite and iOS 8 are released this fall, users of the latest Yosemite beta note. When on a Wi-Fi network where an Apple TV is connected, Yosemite is displaying a message stating that "Apple TV is now available for Continuity," so long as the Apple TV is running new beta firmware. There isn't any indication of what can be done with that support however.
Since the Apple TV has no real equivalents to most OS X and iOS apps, it's speculated that Continuity may allow people to quickly resume music, photos, and/or video on the set-top without having to select AirPlay. If the technology does skip over AirPlay, it would presumably operate through iCloud instead. Current Apple TVs stream media only from iCloud, local iTunes libraries, and AirPlay-enabled devices.