My thermostate refuses to come on. It just says HOld AC..

You could have lost the 24 volts to the thermostat. It might be just working on battery backup.

How do I open my Honeywell Chromotherm III so that I can check the batteries?

Two small screws hold it onto the subbase. They are phillips type and are located on the lower left and right sides in the front of the stat. Loosen the screws all the way and lift out and up from the bottom. The hidden hinge is located at the top of the stat.