There’s a section on podium procedure in the Sporting Regulations (Appendix 3), but not all the requirements appear to be in there – I think some of them are verbally briefed before each podium, according to circumstances.
As I understand it, the drivers are required to accept the trophy, stand respectfully on their assigned podium step for the anthems and then take their trophies and champagne (while meeting other regulations applying to them for the entire event, such as not mixing F1 with politics). The one thing definitely in the regulations is that they can’t be delayed on their route to the podium. The other regulations concern the behaviour of other people involved.
I have no idea who’s handing out trophies, and the winner is likely to have theirs given out by a politician. The president of the ASN will probably hand out 2nd or 3rd place’s trophy, which in my view is not an improvement. The constructor will get theirs from a sponsor’s representative (i.e. Gulf Air), which is somewhat less controversial.
The big thing they can do that is different to normal is they could decline to spray the champagne, since this isn’t a regulatory requirement. A simple “It didn’t feel right” would work as a defence, but attributing it to the protesters wouldn’t. Willingness to do so may depend on what happens between now and then – if there’s some major disaster (connected to the ongoing politics or not), then there would almost certainly be no champagne. Otherwise, it’s more difficult to tell. I could see Webber and the Force India drivers not spraying champagne if they took the podium. Not so sure on the others.