Replace VPI TNT rubber belt with nylon thread

The string shouldn't be overly tight. Just enough to keep the slack off and make the platter spin. too tight and it seems to constrict and over-control the platter. Let the mass of the platter do its thing.

I have the TNT with the single fly wheel. Should I try replacing just the big band or both?

Do both spans. The short length between the motor and the flywheel likes more tension than the platter span. Heavy cotton or poly thread requires a bloodknot; thicker silk requires a loop knot with the knot positioned to travel on the outside of the span. It takes a bit of practice to arrive at the correct tensions, but once there you won't go back.

I own a VPI Scout and my dealer recommended replacing the rubber belt with a silk thread when I was buying it. Since I don't have a speed controller, would you say it'd be pointless to do it and better to stick with the rubber belt?

You can't do it without a speed controller. The different diameter of the string vs belt will result in a major speed difference.

Here's a link to an on-line vendor for inexpensive silk thread in any color you may want. I recommend size or 3.

Griffin Silk Bead Cord. Link below for online supply:
http://www.artbeads.com/stringing-materials-griffin-silk-bead-cord.html