Does the weight of a slingshot frame have any effect on shooting?

It sure can. A heavier frame can have a dampening effect on the shot helping with what you might call recoil.

However, we would warn anyone to not get too hung up on this and think that they need a very heavy frame.

A heavy frame could also be difficult to hold at arms length, so there would be trade-offs on this.

As we mentioned all season long, the trick to so many of these answers is to find what works best for you. If you find that heavy slingshot is difficult to hold firmly, steer clear and choose a lighter frame. If you find that a heavy frame helps stabilize your shot on the release, go for it, and shoot heavier frames. That which works best for you is best for you.