Yes, "Did we finish the jam?" is perfectly natural-sounding.
Did you eat the last of the jam? [if you want to know if that person specifically finished it]
Is there any jam left?
Did we finish off that jam from last week?
Is the jam gone? [but be prepared for Pirates of the Caribbean quotes]
Did we use up the jam?
As J.R. noted, the "use up" phrasing is good for an ingredient - something you buy in a jar or a bag. It generally wouldn't be used for something you cooked.
Did someone use up those potatoes I bought last week?
Did someone finish that potato salad I made Friday?
Are we out of jam? [informal]
Have we run out of jam? [slight less informal]
Do we need to get more jam? [depending on your household, the answer to this could be "yes" even if the existing jar is still half full]