My mom's furnace will not shut off unless I turn off the main breaker. I have checked the thermostat wiring and it is all fine. Also the thermostat is less than a year old. Is there a relay that could be bad causing the problem?

Yup the relay sounds like it got stuck in the open posision or ON position. That or the contactor went bad. You should check them out for conductivity (if you know what you are doing) with a multitester to see which component went bad.

disconnect the red wire at the tstat,, does it (furnace)stop? If not disconnect the red wire in furnace going to tstat does it (furnace)stop this time? You didn`t say if you had a gas or electric furnace,if its gas when you remove the doors it will shut off,so when you disconnect the red tstat wire in furnace you will need to hold in the door safety button to see if disconnecting the hot wire to tstat has helped,, also you need to tell us brand name and model numbers of both the thermostat and the furnace for farther help (( after you have disconnected the tstat= will help determine if problem is in tstat or in your furnace