As troubleshooting home appliances go, the simplest explanation is usually the most likely one. Therefore troubleshooting your oil furnace's furnace controls begins with verifying your thermostat's settings. Make sure the thermostat is on and the system switch is set to "heat" or "auto." Also verify the thermostat's temperature setting and make sure it is set at least 10 degrees Fahrenheit above the air temperature. After changing these settings if they need to be changed, allow five minutes for the oil furnace to respond and turn on.
If your thermostat is battery-powered, the batteries may be drained and will need to be replaced. Similarly, if the thermostat feeds off of the oil furnace's electric power, any interruption to the furnace's power will also affect the thermostat. Check the furnace's circuit at your breaker box to see if the breaker is tripped or if its fuse has blown. Again, allow a few minutes for the oil furnace to respond after restoring power to the furnace.
If the oil furnace itself is suffering a malfunction, it often will not respond to a heating command from the thermostat, leading some homeowners to incorrectly assume that the thermostat has failed when the problem is actually a general furnace failure. Check the furnace's oil supply and replenish it if it has run dry. Also check the furnace's pilot light for any malfunctions and replace it if it has failed.
Of course, it is always possible that the oil furnace's thermostat has simply failed, in which case it will need to be replaced. In this circumstance, it is important to first contact the manufacturer of your oil furnace to make sure that the replacement thermostat you purchase will be compatible with your furnace. Install the replacement thermostat per the manufacturer's directions and test the new thermostat to see if you can now fully control your oil furnace.