Does activating Find My Phone deplete the battery on an iOS device being tracked?
Yes - triggering a location update does use more battery than if the phone were simply idle and chatting from time to time with cell towers to be ready to receive a notification, email or phone call.

No - the high power location drain when you have a GPS mapping program running in the foreground isn't what happens when you use the web site to request one location update.



As long as you are not constantly sending a ping noise or managing to hammer the find my phone interface with a non-stop stream of requests, the phone will be quite efficient with the screen off, a quick location update and then it will go back to idle power and consume no more power than if it were sitting in your pocket.

Think of the function as this:


You log into find my phone (web or another iOS device) and the service looks up all your devices and sends one refresh to each.
The device gets a notification over either of the connections it always attempts to keep open to Apple via both Wi-Fi and cellular data channels.
Apple says - "hey - figure out where you are and then report back"
Your phone does it's location dance as best it can with available GPS, cell tower, and Wi-Fi and then tells Apple where it thinks it might be (and how accurate a fix it thinks it has achieved).
Apple says thanks and your phone goes back to being idle (presumably powering down GPS once not recently requested).
You then see that location update on your display. Apple caches the data, so you will see the last known location for quite a while before it goes to "location unknown" over a period of hours and the last fix becomes stale.