I've never seen the ticket on sale online, so I don't think it is possible to buy it ahead of time.
I've seen a line at the Torre del Mangia, but not super long and not always... so just go by and see how it is, you might then head to the Civic Museum and get the combo ticket there and then return later to the tower.
I just double-checked and want to correct the info I gave above as well as add more details:
I confirm that the Torre del Mangia tower ticket CANNOT be reserved ahead of time, it has to be bought there. Cost is 8 euro. Kids under 6 can climb up free, but of course need to be with an adult (up to 14 years old have to be with adult).
The Combo ticket Torre del Mangia+Civic Museum is 13 euro and HAS to be purchased at the ticket office for the TOWER, it isn't offered at the museum's ticket office (who knows why?).
The ticket office for the Civic Museum instead offered other combo tickets, in case you're interested in visiting other museums in Siena.
There is combo tickets for:
1. City Museums - for Civic Museum and Santa Maria della Scala - Euro 11.00 and valid for 2 days
2. Winter Pass called "Siena Itinerari d'arte invernale": Euro 14.00 and includes the Civic Museum, Santa Maria della Scala, the Museum dell'Opera and Baptistery of San Giovanni - it is valid 7 days
3. Summer Pass called "Siena Itinerari d'arte estivo": Euro 17.00 and in addition to the museums listed for the Winter pass, also includes the churches of Sant'Agostino and San Bernardino - the pass is valid 7 days
If you're in Siena for several days, the passes are definitely worth considering!
I can't select the dates 14, 15 and 16.
Does it mean that the Duomo will be closed then?
I see your earlier discussion here so I am going to purchase the ticket while in Siena not on-line. But I'd like to know if the sights will be available on these dates.
So sorry didn't see your post in time to be of help - the church was most definitely closed on August 15 (day of the Assumption, a religious and national holiday in all of Italy) and for limited times on August 16, as the Palio was being run in Siena. Best bet was to go directly and see how lines were.... hope you can come back and let us know how your visit to Siena was, especially if you ended up there on the day of the Palio!