All a thermostat does is connect the Red (power) to the White (heating, first stage) when it calls for heat. It is effectively a switch that opens and closes (closed should turn on the furnace). But, keep in mind, these colors are what are marked on the thermostat, and they may have used different colored wires in the cable, so be careful. First thing is to make sure the wires are tight in their connections.
The power on the red is (or should be) around 24vac - enough to spark, but normally not enough to hurt you except under really unusual circumstances. With many of these digital thermostats, removing it leaves the mounting plate and electrical connections on the wall. If you were to take an insulated piece of wire and connect (short) the red to the white, the furnace should come on. If it does, then it is the thermostat. If it does not, then it is the furnace. If you don't feel comfortable doing this, then you need to call in a qualified service man.
If it ends up being the furnace, it could be numerous things, but that's another whole troubleshooting sequence. Your operator manual for the furnace should help there.