Will Elizabeth warn me if I rely on her in combat too much?
In one of the original trailers, Elizabeth conjured a storm cloud, which Booker then electrified to damage everyone nearby. Afterward, Elizabeth was exhausted.
Shortly after seeing that, I remember reading a developer preview article which said that too much use of Elizabeth's tears would start to weaken her.
Right now, I've just finished the Hall of Heroes and I haven't seen any evidence of that at all. In the past few combat sequences, there's been just tons of things that I can ask Elizabeth to conjure up for me - mechanical patriots, barrels of guns, health kits, machine gun turrets, and so on.
I haven't taken advantage of many of these (only the first time when she offered to pull through some cargo hooks or a heavy machine gun turret and a couple of health packs since) because I don't want to hurt her. So far, I haven't seen any evidence of damage, but now some of these fights are getting tough without her help.
Will Elizabeth warn me if I start to use her powers in combat too much, or will she just suffer silently?
Elizabeth doesn't weaken from opening tears. Also, Elizabeth doesn't get damaged by fighting at all. You don't need to worry about Elizabeth while fighting (a loading screen hint mentions this). About her loot giving: Elizabeth will tirelessly give you loot subject to different conditions. (More details in this question: How does Elizabeth getting various loot and then giving it to you work?)
In one of the original trailers, Elizabeth conjured a storm cloud, which Booker then electrified to damage everyone nearby. Afterward, Elizabeth was exhausted.
Shortly after seeing that, I remember reading a developer preview article which said that too much use of Elizabeth's tears would start to weaken her.
Spoilers:
We were deceived by the False Prophets at Irrational Games. None of those things from the trailers were actually in the released game.
(Or maybe we weren't deceived at all...)
You can open tears as often as you please. Elizabeth will not tire out or be adversely affected.