No, it is not necessary for a user who wishes to take part to have a Facebook account.
The administrator can decide whether or not users should be obliged to identify themselves via a Facebook account. In this case, at the moment of participation, the application will ask the user to begin a session in Facebook and accept the application permissions:
In the case of not requiring users to log in to Facebook in order to participate, users can complete the registration process without needing to be connected to Facebook.
Note: In this last case, we recommend that you always ask for the email in the registration form and that you enable the connection status bar. Here we explain how to enable the connection status bar.