I have hot water heat in my house and a programmable thermostat.
The pump and boiler will come on to heat my house, but will stay on even after
The pre-set temp has been reached. When I try to manually override (using the thermostat), the boiler and pump stay on--even if I shut off the system using the thermostat. I finally have had to use the breaker to make the system shut off.
But when I turn the system back on (the next morning), the system will work the way it should for a while, but then it goes back to staying on and not obeying the thermostat.
I replaced the thermostat with a brand new one--the problem stays the same.

Sounds like you have a relay sticking or possibly a short in the thermostat wiring. Try disconnecting the thermostat wires from the boiler. If the boiler & circulator shut down, it is likely a problem in the thermostat wire. If it doesn't the problem likely lies in the boiler control.

Many boiler systems are wired off the main burner control to maintain the boiler water temperature. In other words the boiler will stay at lets say 160-180 degrees at all times weather the thermostat is calling for heat or not. This is usually done on boiler systems that also supply domestic hot water along with hot water for heating. So you might have to check the boiler control or (Aqua-stat) as some are called to see how it is wired or test the control system to see if that is the cause of your problem/situation.

A pinched or shorted thermostat wire can also cause this.

Aquastat - Wikipedia, the free encyclopedia