Speaking here from very little experience with 4WD, but with enough book knowledge to get myself in trouble. So be gentle....
With an AWD vehicle (which is what 4x4-Auto in the Explorer really is), some torque (35% IIRC) is always going to the front. As long as there is no slip front to rear, the viscous coupling is not getting used, and heat will not build up. The problem is when 1 wheel has to turn faster due to smaller diameter. Like someone said, with open diffs, it's probably not an issue. But if you have a limited slip axle, I think it could introduce a problem.
I remember reading that if heat build in the viscous coupling to the point something will be damaged, the center diff will automatically lock until the temperatures go back down. For my Mounty, the 4x4 indicated lights in the dash. For the Explorer, that would have the 4x4-Hi engaged. When locked, there is no slip, and the coupling can cool back down.
So that is probably the answer. Get the bad tire replaced. If you find yourself going into 4x4-Hi without pushing the button, you probably need to replace the others.