My furnace thermostat (10 years old) has suddenly decided that it will no longer start the furnace in what's called "hold" mode. It only works in a previously set "program" mode. I am able to change the times and temperatures in "program" mode, but the furnace will not even start/run when I change the thermostat to "hold" mode. Anybody have any ideas?

Stats aren't that expensive and there are few/none repairable parts in electronic units. Some units require a battery while others draw from the furnace circuit board. If you have a battery, replace it and see if it has improved.