This is a pretty dumb question, but are there any legal restrictions on the caliber of a weapon citizens in the US are allowed to own?
I am curious because if there is not such a limit, couldn't someone buy one of those surplus Main Battle Tanks, and use non-explosive rounds, or would it be a destructive device with a $200 per round tax on each shell?

Yes. Federal involves "modern" weapons posessing a bore diameter larger than 1/2 inch .
A 200.00 tax stamp is required.  Also, each explosive shell requires a 200.00 stamp.

The cost of maintaining a modern MBT, (Say a Chieftan) is trememdous, let alone transporting it, or finding a safe place to actually fire it's weaponry.
However, there ARE collectors and firers of artillery. Ones I've personally seen are 20mm Lahti ATG's, a few 25mm Puteaux, and a PAK 38..

There was a nice WWI vintage French 75 for sale a while back too.
The limiting factors seem to be finding a range..

Meplat-

Okay, so if the weapon has the over .50 Caliber bore, there is a tax stamp, but are there extra tax stamps for non-explosive shells? such as 20mm?

only shells containing more than .5 oz. of incendiary or explosive compounds, so a solid bullet I don't think would be a DD.  I have live 30mm target practice rounds from an ammo dealer and there was no paper work for the above reason.