Installed password protect this afternoon for my clients and one immediately had an issue. When putting in a password, it would just go back to asking for the password.
I switched themes back to twenty-11 and still no problem. I then started disabling plugins looking for a conflict til I got to the point where only password protect and domain mapping were active. No joy. I then disabled domain mapping and put in the password. At that point, it took me to the admin login screen with the original (non-mapped domain). I clicked back on the browser and refreshed the page under the original domain and all the protected content was there.
I then tested the plugin with all other plugins reactivated on a non domain-mapped site and it worked like a charm, but still not on any of the domain-mapped sites.
So it would appear the issue lies with the domain mapping plugin which is also your (wpmudev) plugin.