Why would a blower motor work with A/C, but burn out with furnace?
Lots of dust and debris on the vents means your air filters pass dirt.  2" pleated MERV 7-8 are good if your system will handle thicker media.  1" pleated are OK but may be more restrictive.  In my region, specialty filter supply houses are the best place to purchase filters.  They will sell retail at a fair price, and if you need a nonstandard size they will make a custom fit for a small fee.

High heat causes motor insulation to fail prematurely.  If the motor windings are burning out frequently during heating season and not cooling season, the cause is probably that the high heat is worse during the heating season when the blower is delivering hot air.

I can think of four possible causes:


The blower capacitor is weak causing the motor to overheat (a good tech would usually replace both the motor and the capacitor after a motor failure).
The motor is undersized.  The HP and RPM of the motor must match the fan.  If someone replaced the blower motor with an incorrect unit on the first go-around, then new motors that size will continue failing.  Certain Goodman units have a blower that requires an uncommon lower RPM.  On this unit, the typical 1075 RPM 1/2 HP motor is not an acceptable replacement and will fail.  Checking the motor amperage against the nameplate would expose this problem.
The air temperature from the furnace is too hot.
The voltage supplied to the blower exceeds the motor nameplate rating by more than 10%.