Furnace blower motor won't turn off
I have learned to not dispute things Tester has said, it is bad for one's reputation in this part of the world. I will not say he/she is wrong in this answer.
I will not be insulted if you just pass by this answer.
--- WARNING ---
I write too many words and use too much detail in my writing, always have and always will. Even my warning was too long.
Having said that, I offer this simply as data to show how Tester could have reached his/her conclusion and provide some background information. Hopefully it will help others looking for detail.
If anything I write seems useful, but confusing, let me know, and I will do my best to explain my thoughts. I am not an ass, it just seems like that sometimes.
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Home HVAC control is more complicated than it needs to be, this is because as features were added - Air Conditioning, Heat Pumps, 2 stage heat, more control wires were added to the system.
The advantage is that everything is backwards compatible, clear back to the day that thermostats were invented.
A couple of disadvantages are that some wires are now redundant and there is not a standard for wire colors.
The color code became an issue, because when what was then 'standard' thermostat cable ran out of wires, the solution was often to pull another standard cable. This results in 2 of every wire color. This is where things stop being simple.
Terminal Designations and -- 'conventional' wire colors, don't place bets on them being right --
R - Power - 24 volts, to the thermostat - like the Hot wire in a light circuit - Red
RC - Cooling power -  Air conditioning power into the thermostat - Red
RH - Heating power - heat power into the thermostat - can be the same wire as RC - Red
Y - Air conditioning compressor power - this is the piece outside of the house - Yellow
Y2 - Second stage A/C compressor power - not all compressors have this feature - Light Blue
W - Heating terminal - Heat power out of the thermostat - White
W2 - Second stage heat - Not in all furnaces, for Heat pumps this is Emergency Heat - Brown
G - Fan Power - Green
C - Power to thermostat - this is similar to the Neutral wire in a light circuit - Black
O or B - Power to heat pump Reversing Valve in outside unit - Orange or Dark Blue
E - Emergency heating on heat pump, a different way to get emergency heat - None
X or Aux - Back-up power/auxiliary, another different way to get back-up heat - None
S1 &amp; S2 or Outdoor1 and Outdoor2 - Temperature Sensors, not connected to thermostat wires

Now, to Charlie's question:
I read '... continues to heat the house ...' as the furnace stops heating when expected, and not that is always continues to provide heat. This is confirmed with the statement ... heat when called for ... .'
First, remove the wire on the 'G' terminal of the controller in the furnace.



If the fan stops, the problem is between the wire that was on the 'G' terminal out to the thermostat and the and back to low voltage (24 volts) transformer located inside the furnace.




If the fan keeps running, the problem is from the 'G' terminal on the controller to the transformer. See step 5.



-- 3, and the fan stopped, attach the G wire back at the controller in the furnace. Expected result is that the fan will start again. At this point, go to the thermostat, and remove the wire from the G terminal. Move to step 4
-- 4, If the fan stops the problem is a short on the wire that was attached to the G terminal on the thermostat and the wire that was on the G terminal on the controller in the furnace. Most likely a pinch from something. It will most likely to have happened where the wire is tight. I have found problems in my system just fractions of an inch from the screw terminal in the thermostat.
This will be a short, and not a broken wire, a broken wire would cause something to stop, and not continue running. Since the short seems to only be between the G wire, and a power wire, expect it to be somewhere that the wires are not inside the jacket of the cable.
This is because it would be unlikely for the short to have happened on the jacketed segment, and not seem to have effected any other wires.
Crushing the entire jacket would likely, but not certainly, short more than the 2 wires. If you find the short, either separate the wires, and some tape on them, or use another wire, if you have a spare in the cable.
Hopefully you can find the problem in this 1 one run of cable, and you will live happily ever after.
-- 5. Remove the wires, after marking them, one at a time, starting with red ones, then black, from the controller terminals. If you remove a wire, and the fan stops. The problem is from that terminal on the controller to the transformer.
If all the low voltage wires are removed, the problem is either the controller, or somewhere between the fan harness and the fan. If you disconnect the fan harness from the controller and it continues to run, the problem is a short to ground from the harness to the motor. Or, and this is unlikely, demonic possession. Unless there are many wires, more than than three, then you have a state of the art, electronically controlled fan motor. The problem could be either the motor, or the 'module'. Replace the both parts, you will thank me later.
If the fan stopped when it was removed from the controller, the problem is in the controller. Replace, reattach all the wires, carefully, and live happily ever after.
--- If it is still not working. Then it is time to start drinking.
But seriously, let me know and we will get serious about fixing this thing.
Best of luck to you.