It is OK to keep the system running for 20+ minutes. But if you leave the stat in heat on position all the time, the system will remain on all the time. Your house will become an oven. Normally the system turns on intermittently -- it only needs to run when the temperature drops below the programmed value.
It sounds like there is something funny with the thermalstat. You chould probably buy a simple one at Homedepot for $20 and install it by yourself. The control wires are color coded. I remember there were only 6 or 7 wires. Please read the installation manual included in the new stat first. Then open up the old stat, take a note of the existing wiring. When I installed my super-duper progammable stat, I actually put labels on all the existing wires first before I unsrewed them. It was a very small DIY job.