Ok, here is what it sounds like is going on.
IF the heat exchanger in the furnace gets too hot, a hi limit sensor shuts off the main burner AND energizes the blower relay on the control board to keep your fan running until the heat exchanger has returned to a safe temperature.
The hi limit control looks something like this:
It is located in the front section of the furnace behind the gas valve.
The problem is usually a lack of air flow through the furnace.
This can be caused by a dirty furnace filter, too many vents shut off in the house, or a dirty A coil. (The A coil is the A/C coil that sits on top of the furnace that provides cold air in cooling mode, assuming you have central air). Over the years dirt and pet hair (if you have one) can accumulate on the bottom side of the coil and reduce air flow to the point where the high limit acts up.
Check the airflow out of your vents to see if you have a strong airflow or seems weak, you have to get your hand close to vent to feel airflow.
The other possibilities that cause this could be a slight crack or hole in the heat exchanger causing flue gas to go into the air stream around the limit sensor. This is a serious problem and should be checked out by a tech. One way to tell if you have a crack is to look at the burner flames on your furnace. You should have stable blue flames shooting into the heat exchanger. If one or more of them suddenly change, burn back or become disturbed when the main blower kicks in you probably have a problem in the heat exchanger and need to have it checked immediately.
Last, you may simply have a defective hi limit sensor. For testing purposes ONLY, you can remove the two wires, (usually red) and temporarily hook them together to see if it turns the fan off. Although it is only 24 volts going to the wires, make sure power is off if you try jumping them and make sure your jumper doesn't touch any metal in the furnace or you'll likely cause more problems.
If you are not comfortable trying any of these suggestions, by all means get a qualified furnace guy back out there. A hi limit fault is serious as it indicates an overheat condition and needs to be rectified.
Check the simple stuff first. (filter, airflow).
I hope it turns out to be minor. Feel free to reply with follow up questions. Thanks.