Yes. If we're being brutally direct:
The Italian Campaign did divert German troops from both Western Europe, and the Eastern front (indeed, the invasion of Sicily arguably is what caused the Germans to break off the fighting at Kursk).
While Italy is not often seen as a powerhouse, it was arguably the biggest and one of the more capable allies (if not the biggest and most capable!) Germany had in Europe. While elements of the Italian military fought on, it was a shadow of the forces that had been available prior.
Invading Italy in so many words opened up the Mediterranean with few remaining true barriers to Allied shipping (while the Italian Navy and Air Force hadn't been supremely effective, it was still a threat, and a lot of airbases the Germans would have loved to have used against Allied shipping fell into Allied hands).
There's some other "softer" ideas, like it discouraged Spain from entering the war, eroded the overall ability of the Germans to continue the fight etc, but drawing away much needed German forces from elsewhere, knocking out Germany's biggest allies, and making the Mediterranean safer for shipping were all things accomplished by invading Italy.
The question of course, is were those things worth the blood and treasure spent on the Italian campaign. The campaign had been promised in so many words as a route into the "soft underbelly" of Europe, and the resulting often brutal fighting up the peninsula most certainly was anything but soft.