Ok I think I have a clue.. The clock is an automotive type (Dash Clock) and it runs on 12 volt DC, not 120volt A/C.
SO DOES THE FURNACE
When that blower kicks in the draw on the 12 volt system is kind if impressive, IF the battery terminals are a tad in need of cleaning, Or if the battery is down at all, (of course plugged in one would assume it's full up) if the connections to the converter are tarnished.. It can happen the voltage will drop (Can happen.. WILL HAPPEN) and it may drop below the clock's cut off voltage, causing the BEEP, and weird action.
If you can confirm this (use an analog multimeter and watch the 12 volt line as the furnace kicks in) I have assorted cures.
First though, Clean the battery connections. IF that does not fix it, PM me.