Heater doesnt kick on below 70 degrees. When it does kick on with thermostat set higher than 70 it will turn on run and then blow. (Pilot light works) Doesn't blow as long as it should then shuts off and idles (runs) again. Had a certified furnace repair person look at it. The limit switch was replaced. Continued with the same problems. Called the company again, they told me I needed the thermostat replaced (bought my own and had it replaced) continues with the same problems. $300 later and heater still doesn't work, company won't guarantee their work because I bought my own new thermostat instead of buying their expensive one. What now??

Make and age of furnace? If it uses a fan/limit control I would start there.