I don't think so,
in fact it should reduce beltwear.
since the kart will be in a lower ratio for a longer time, it can accelerate more easily and overcome road inclination mass inertia and stuff;
also the driven is grabbing harder on the belt
(and slip is what wears the belt, not friction - just like car tyres

)
Anyways.. going off road will cause many things to increase beltwear
starting from higher rolling resistances on dirt than you'd have on tarmac,
potholes act like small hills that have to be climed (road inclination) etc.pp.
So, it's better to have it at three than at 2 if you can make use of the additional torque
(or more precise increased torque-time

)
but I'd say it'll still wear faster than on a kart meant to ride on asphalt
'sid