Hmm… I usually drop the “Dear Professor” (or most usually “Hi Professor” or "Bonjour Professor) after a couple replies, as does the professor. So it looks like:
Bonjour Professor,
I’m going to be applying for a study abroad scholarship. Could you please write me a letter of rec?
Merci,
harvestmoon
Bonjour harvestmoon,
Absolutely. When is it due?
Madame Professor
Bonjour Madame,
It’s due in 3 weeks and here are the forms.
Thanks so much!
harvestmoon
Merci, harvestmoon! I’ll get it done within the week.
Madame Professor
There’s just a sort of natural drop that happens. I will say that I usually err on the side of formal and wait for them to become less formal, but still, for me it isn’t a matter of taking too much effort to write “Bonjour Madame” every email as it is too formal. A French friend of mine said she avoids using “vous” unless she absolutely has to because she thinks formality creates unnecessary distance between people, and I kind of feel the same way here with salutations.
I guess all of this does depend on your relationship with the professor, though. I have some where I can be like “hey first name, write me a rec please! :)”