In reply to Roberttaylor:
We were there the first two weeks of October (2018) and it was a perfect trip. Eleven days in the Dolomites and it only rained for one of those. That day was forecast well in advance (I found YR to be spot on) so we planned it as a rest day. All other days were blue skies and cool temps.
All tourist infrastructure seems to be open until 31st September. Almost all major lifts seemed to be open until 15th October. Campsites, refuigios and restaurants do start to close from the end of September but nothing that you can't plan around. For example, we climbed on the Vajolet towers and while Refugios Alberto and Vajolet had closed for overnight stays, Refugio Gardecca was still open. Several of the campsites and lifts we used closed October 15th so I suspect late October would start to become trickier with planning.
The benefit was having the classic routes completely to ourselves along with accommodation being cheap, roads being quiet and weather being perfect. Any downsides of shorter days and some restaurants and campsites being closed were easily worthwhile.
If I was to go back I would almost definitely plan to do so between mid September and early/mid October.