The legislation applies to bureaucrats, police officers, teachers, and bus drivers, as well as doctors, midwives and dentists who work in publicly-funded hospitals or clinics.
Exceptions, however, can be made on a case by case basis.
"I should see your face, and you should see mine," Quebec Premier Philippe Couillard said of Bill 62.
Speaking to reporters, he cited improved "communication, identification and safety" as reasons for the ban.
But critics say it will effectively marginalize them by limiting their access to government jobs and services.
The National Council of Canadian Muslims said in a statement the law "boils down to ugly identity politics" ahead of a general election in Quebec next year.