25.4e of the 2013 Sporting Regulations states:
At the start of the race each car which took part in Q3 must be fitted with the tyres with which the driver set his grid time. This will only be necessary if dry-weather tyres were used to set the grid time and if dry-weather are used at the start of the race.
Any such tyres damaged during Q3 will be inspected by the FIA technical delegate who will decide, at his absolute discretion, whether any may be replaced and, if so, which tyres they should be replaced with.
A penalty under Article 16.3(b) will be imposed on any driver whose car is not fitted with the tyres with which he set his grid time (except if damaged tyres have been replaced with the approval of the FIA technical delegate).
So if a driver has a puncture after his hot lap then that set may be replaced for the race.
Thank you for that. Do they get additional set?
Does the circumstances of how the tyres got damaged matter? If you clipped someone and punctured your tyres is it different from driving over debris and puncturing your tyres?
As far as I can tell that is the only regulation covering the situation and the crucial part is that what happens is entirely at the technical delegate's discretion, which means that there are no definitive answers to your questions: it would be decided by the technical delegate on a case by case basis.
As a reference point, Hamilton had a puncture during qualifying in Singapore in 2011 and he was allowed to replace the tyres but not provided an additional set. http://www.formula1.com/news/features/2011/9/12579.html
. So my understanding is that they don't get an additional set. This is consistent with the fact that if a team has a puncture during the race they don't get an additional set and the tyres used in Q3 are one of the sets used in the race, so there is no real distinction. In terms of advantage/disadvantage it cuts both ways: the driver starts on fresh tyres, but they have one less set for the race.
Whether the circumstances of how the tyre came to be damaged play a part in the decision IMO this is where the technical delegate's discretion would most significantly come into play. There's no specific reason why driving into someone would be regarded differently to driving over debris. My guess is that the only time the technical delegate would refuse is if there was some indication that the action was deliberate in order to get a fresh set of tyres, in which case the driver would probably be disqualified from qualifying and sent to the back of the grid.