As always it depends what the weather is doing. Be mindful that the temperature starts to drop and most guiding companies end their trips towards the end of September - this should tell you something. We summited in the second week of Sept 2010, minus 25C on the top. Be prepared, stay safe.
In reply to llanberis36: summited 14 Sept 2011, conditions v good. It was quite cold but not unbearable. Gouiter hut was quiet. Only about 25 people summited from our hut so felt good not to be in a crowd.
We spent 2 weeks in Cham and had an open mind re the route, the weather went our way so after 10 days acclimatisation we managed it.
I was up the Gouter on 26 Sept this year. Nice, but windy and cold on summmit ridge and not sure anybody got to the actual top of MB. Lots of returning punters. But don't think it was any colder than June or July when I've been in the area previously. The Gouter Hut closed at the end of that weekend.
I've been to the top of the Dome de Gouter in mid-September (after doing the north face of the Bionassasy) en route to the Gouter hut. It was mild and still, with excellent snow conditions. That time of year can give some of the best settled weather of the year: it doesn't get so hot so you don't get afternoon storms. I've done several 4,000-ers in September and some big routes. However, it can also be a washout: I've experienced that too.
It doesn't necessarily follow that September is automatically colder at altitude than July August. This photo was taken around the 25 September a few years ago. http://www.ukclimbing.com/images/dbpage.html?id=233018 That day we summited wearing a light fleece over that thin shirt. Equally I've been on the summit of MB quite a few times during 'the season' and been so cold that I've worried that if the slightest thing went wrong, then I wouldn't be able to cope. My advice is to go for it but don't hesitate to take as many extra layers of insulation with you as you can carry if low temps are forecast. A really good tip is to take a pair of mitts - insulated and loose fitting - they don't have to be expensive - you can probably find some for a few €uros in Chamonix.
I climbed it a number of years ago in the second week of Sep in perfect weather. It was my first alpine route. We had been unsuccessful in an attempt on the Monch in the previous week due to bad weather. Not wasted time as it helped with acclimatization (we spent a night at altitude).
We aimed to go up by the Gouter route and down by the Grands Mulets route. The Gouter route was in good condition, but the Grands Mulets route was not in condition with thin snow covering the crevasses, so we ended up climbing back up and descending by the Gouter route (knackering!). It might be that the GM route is not a good option late in the season - probably worth investigating. You can check current conditions of the routes with the guides office in Chamonix. They are very helpful.
If you're thinking of using the telepheriques, you may want to check what dates they stop. We went to Chamonix for rock climbing in mid-September (on the 18th I think) a couple of years back and found that nearly all of the lifts had stopped for the season on that day! Having said that walking up from the valley is a nice way to climb Mont Blanc.
The Aiguille de Midi lift runs later than the others but is expensive!