Thanks for the suggestions. I found that there is indeed an led light
on the motor housing. And to my surprise, when I got up on a ladder
to watch the led light while I pressed the remote button, I found that
it worked to raise and lower the door. Essentially, it seems to be a
distance problem. If I hold the remote within a couple feet of the
receiver, it works. Any further away than that, and I get no response
from the receiver or led light.

Any new ideas? When/why would a remote only work at such a short
distance? I'll try cleaning the contacts inside the remote, but they
look pretty good. Maybe the replacement battery I purchased was old
and near-dead? Hmmm...