I work as a teacher in France, where removing the Hijab is a requirement for teachers. In this case, would my salary be considered allowed since I am being paid for my teaching skills rather than my physical appearance?
You as a Muslim female is responsible to obey the rules of Allah including wearing full Hijab where ever you can be seen by any non Mahram (outsider) man. This responsibility is more important than doing any specific job which prevents you from obeying Allah, as far as there are other jobs which don't prevent you.
If you teach in a girls school where no man can see you with out Hijab, then it is permissible because you don't need to wear Hijab among females, but you must be sure that no man at all can see you with out Hijab.
If the job exposes you to men and forces you not to wear Hijab, you should look for another job which does not prevent you from obeying Allah (SWT).
Wassalam.