Does the thermostat has an 'anticipation' adjustment? Is the thermostat smart enough to understand the various types of heating (baseboard, forced hot air, radiant, etc,)? That needs to be set for the type you have. Depending on the thermostat, you might be able to hear a relay activate when you increase the setting above the current setting. If the thermostat is sitting in direct sunlight, or in front of a heating duct or light fixture, it might actually not be sensing the room temperature.
A unit could have a timer that forces it to not restart after running for a certain period of time, and that may need adjustment. If you're waiting to hear the blower fan come on, keep in mind, many furnaces will let the burner run for a bit prior to turning the fan on. Have you had someone check to see if turning the thermostat up actually starts the ignition sequence? Depending on the type of unit, that might take up to a minute or so, especially if you tried this in quick succession...with say a gas furnace, it often will wait a bit for any raw gas to dissipate prior to trying to restart.