Does it hurt the transmission to drive Dodge Ram 1500 2001 when the speed sensor is bad?

It could lead to excess RPMs before shifting... these transmissions are electronically controlled, rather than mechanical, but if you haven't experienced it, then the shift points aren't dependent on the speed sensor, and there shouldn't be a problem, either for the transmission or the engine... it just means your speedometer will be out of whack.

If you have noticed such problems, then your shift points are dependent on the signal the ECM receives from the VSS, and, while it won't be an issue so much for the transmission, the engine could be at risk, unless it's programmed with a safeguard to prevent excess revving.

Either way, the VSS is a quick and easy component to replace.