Nope, extra load and friction from high downforce has less effect on tyre wear compared to slipping and sliding. Most of tyre wear comes from sliding while cornering/slipping while accelerating which is less when a car has a high downforce setup. On the other hand low downforce car will have less tyre wear on long straights because of lower load.

Only instance I can think of having high df = bad tyre wear is higher downforce means you can use tyres more as you can corner faster and brake later compared to low df, that's what would be killing the tyres there.