Hello. Our heating unit (inside unit) will not shut off once the desired temperature is met in the house. It isn't the thermostat because it says it's shutting off, but it really doesn't. We've had someone at our house today to look at it, and they say the computer board is fried. Does that sound normal? We cannot leave the heat on for a long period of time, because it roasts us out of the house. Thanks in advance. Oh yeah, it's a York that was suppose to be "new" when the house was purchased. The technician today said the serial number is one from the 80's. Any advice?

Disconnect the white wire from the thermostat and the furnace should shut down, if not then the problem is most likely in the furnace, Good luck ,Mike

He is telling you what is wrong.
He is there and I am not.
I would feel that if he replaces the board and the furnace worked correctly he has done his job correctly.
Now if the board does not fix the furnace he can remove the board and you will not own him anything.
It is a no brainer roll the dice and have him fix the furnace.