Following a power outage yesterday, my remote control celing fan (an Emerson, I believe) stopped working. I replaced the battery in the remote, but still no dice. What can I do to make this fan operable?

Probably a power surge damaged the receiver. Confirm the manufacturer of the fan and remote.

I have a Harbor Breeze ceiling fan with remote. I have to replace the battery at least once a week. I'm sure this is not normal, what could the problem be?

I have no idea, it is something wrong with the remote, you should replace the remote.