I am Indonesian. This month I will go to Iasi, Romania for student exchange. But I will go there after staying one night in France, then take a plane to Iasi, Romania. Do I need to apply for a Romanian visa to enter and stay there for approximately 4 weeks if I already have a Schengen visa issued by France's embassy?
It depends on the exact type of French/Schengen visa you have. With a French residence permit or long-stay visa, it's not a problem. With a Schengen visa issued by France, it's possible if the visa has two entries (or is a multiple-entry visa) and is still valid for the time you would be in Romania. Finally, if you have a single entry visa or a visa expiring during your stay in Romania, then I think you might need a Romanian visa (my reading of the rules is that transiting for 5 days might still be possible but I am not sure).
See also the official rules.
Romania, since Jan 1st, 2014, is now also part of Schengen, hence, your visa is also valid for Romania and there is no need to for a multiple entry visa, you never leave Schengen.
@Relaxed is wrong, since this just recently changed.
edit: Sorry, I was wrong, they were supposed to enter by Jan 1st, but they haven't yet, however, it appears you don't need a different visa to enter/exit Romania.