I've been working on an itinerary for Germany and Austria in December and it looks like we may be in Salzburg on Christmas Day (or perhaps just a partial day as we might be switching from Salzburg to Munich that day - I assume trains will be on a holiday schedule).

I've been in Munich on Christmas Day and know what to expect regarding closures, etc. Is the situation in Salzburg similar?

Well at least churches will be open and probably playing Silent Night - I heard it sung once in Munich's main church on Christmas Eve and it was surreal. The song originates from the Salzburg area.

Looks like the Christmas Markets will be open in Salzburg on Christmas Day, whereas IME most close around 2 pm on Christmas Eve in Munich.

Yes, it should be pretty similar to what you experienced in Munich.
Shops will be closed. Some restaurants may be closed, but not in the touristy areas of Salzburg. Sights are hit and miss, you should check individually.
Trains and public transport will run on regular Sunday schedules - which is not much different from weekday services between Salzburg and Munich. Usually this only affects very early commuter services, like at 4 or 5 am.
Christmas Eve is the big thing in Austria and Germany - but December 25th is more or less like a regular Sunday.

Thanks Cowboy, that helps.