A character protected by the Eyrie is ineligible to be chosen any time a player must choose and kill a character (military claim, Shagga, etc.). For this reason, you can use the Eyrie to choose an opponent's inexpensive character, like Winterfell Steward. Then that character "cannot be killed" so it cannot be chosen for military claim, and your opponent may have to kill something more impactful.
"Cannot be killed" is not a save effect, so the Eyrie will protect from effects that say "Cannot be saved". For example, use this ability when the Plot phase begins (if you plan to reveal Wildfire Assault, or expect your opponent to reveal it). Then you can choose up to three characters, and kill all others except the one protected by The Eyrie, so you'll end up with 4 characters on the board.
- "Until the end of the phase" lasting effects wear off before "At the end of the phase" delayed effects trigger, so The Eyrie will not protect a character from being killed by Tears of Lys. The "until the end of the phase, that character cannot be killed" from The Eyrie is a lasting effect. It creates a condition that continuously applies until a future time or game occurrence takes place - at which point, it stops. But the "at the end of the phase, remove token and kill character" from Tears is NOT a lasting effect. It doesn't actually do anything or affect the game in any way until a future time or game occurrence - at which time, it resolves. This makes the "at the end of the phase, kill..." part of Tears a delayed effect, not a lasting effect. The entry for "Delayed Effects" on p. 6 of the RR says that "Delayed effects resolve automatically and immediately AFTER their timing point/future condition occurs, before reactions to that moment may be used." So, since delayed effects happen "after" their timing point/future condition, the same reasoning for "until the end of the phase" lasting effects expiring before "at the end of the phase" effects can be resolved will apply.