I did a search on here and also looked at Rick’s Venice guidebook first and I cannot pin down the answer to these questions. Thanks.
http://palazzoducale.visitmuve.it/en/regolamento-di-visita/
Click on the English option. Scroll down and you will see the answer is no.
Photos are also forbidden in St Mark's:
http://www.basilicasanmarco.it/informazioni-per-i-turisti/orari-di-apertura/?lang=en
Unfortunately, many visitors don't speak English or Italian, so at least for St Marks, it is widely ignored.
Thanks everyone for the info and links. I guess I can buy photos at the stores. Those will be better pics than I could take anyway.
Sam, even if a visitor does not understand signs in English or Italian, a sign with an image of a camera with a big red "X" on it cannot be misunderstood. Some people, entitled people who think rules are for other people, insist on their right (God-given apparently) to photograph anything, anywhere, at any time.
P.S. I remember when we were crossing the border between two of the Balkan countries (can't remember which border crossing) our tour guide told us all NOT to take photos from the bus window when she went into the office with our passports. Of course, one man decided the rule was not for him and took photos .... with his camera flash on, that is how stupid he was. Anyway, he was escorted off the bus, interrogated for about 20 minutes and had his camera confiscated. The rules were for him, after all.
The "No Photo" is enforced in Basilica San Marco. There were several men enforcing the rule a couple of months ago. The church museum also has the rule but no one was enforcing it.
No photos in the church. But two years ago I took pictures all over the inside of the Doge's Palace and on a tour with no problems.