You leave it running to continue lubrication to the rear bearings in the tranny.
Funny thing, up until a second ago I thought you were proudly talking about your all terrain tires
If your truck has some sort of AWD/part time system and you tow it on a dolly without removing the rear DS you risk damaging the coupler.
If your truck does not have some sort of AWD system and you put it on a dolly but throw the tcase in neutral, and do not remove the rear DS you will be spinning the front DS... Potential issue there..
Funny thing, up until a second ago I thought you were proudly talking about your all terrain tires

If your truck has some sort of AWD/part time system and you tow it on a dolly without removing the rear DS you risk damaging the coupler.
If your truck does not have some sort of AWD system and you put it on a dolly but throw the tcase in neutral, and do not remove the rear DS you will be spinning the front DS... Potential issue there..