If you only have 2 wires going to your t stat then you wont be able to turn just the blower on.
Does the blower shut down at the same time as the burner?
There likely is no way for the unit to detect the thermal cutout in the blower, also a blower is self cooling so it overheating is not common.
More then likely it's a heat exchanger over temp (high limit switch) trip. This is often caused by lack of airflow through the heat exchanger.
A few things to check.....Make sure all your room registers are open, check the filter. Look for a forgotten about second filter (only need one). If the furnace was run without a filter, the AC evap or secondary heat exchanger (if you have either) could be plugged up and may need a professional cleaning. Also, if your blower requires oil every few years and the bearings went dry they could be damaged which can slow the unit down which will raise your heat exchanger temp. If you haven't oiled the bearings and they are due, add oil. (refer to your manual) If the blower is making noises when spun by hand or while it's spinning up/down then the bearings may be going.
If you look at the terminal strip where your t stat wires connect to the furnace (may be under a metal cover, disconnect AC power first), you may see a terminal marked G, or have a green wire on it.. that is your fan signal.. jump from red (should have a wire going to your t stat connected) to green then connect AC power.. the blower should run. If it runs and stays running then this will confirm your blower is good.