Hi,
We just moved into our new rental. Our bedroom has a harbor breeze fan with handheld remote. The fan starts fine and everything seems to work, but sometimes the fan will run for a short time and then shut itself off, and other times it will run all night. Could this fan have a timer, and if so, how do you use the remote so that it will run all night? We obviously don't have access to the user manual for this fan, so any help would be great!

I would change code in remote and fan. Neighbor? Maybe on same code.
If no automation boxes or equipment, Not likely, but remote could be made with timer.
After you change codes and turn on the fan, remove the batteries from remote, to verify
Your remote not the cause, could be intermittant switch on remote.

You need to change the frequency of the fan and remote. See:

Ceiling fan remote controls - help and FAQ- Ceiling Fans N More