Fiction and history are full of accounts of people riding horses until they drop dead. I've been asked before - can that actually happen? Will they let you do that?
The answer is: Yes.
Horses are very smart animals in some ways - and not smart at all in others. A domestic horse that trusts humans will let you ride it to exhaustion with only minor complaints. (Mules are a different matter - a mule is much more likely to refuse to let you overwork it, put too much weight on it, etc). It's up to the human member of the partnership to watch for the signs and know when to stop. Additionally, a panicked horse will run itself to exhaustion. In an emergency, the horse will often pick up on the rider's panic and run extra hard.
So, it's perfectly feasible to have somebody ride a horse to death in your story - assuming the situation is such that they don't care (or can't afford to care) about the health of the animal.
Not a pleasant thought, but there it is.
(This horse, on the other hand, just enjoyed a multi-hour ride, and while she's covered in sweat, look at those ears).