We've covered Microsoft's push into the realm of two-step authentication quite extensively as the company continues to roll changes out and patch up broken areas of its backend. This includes the Dev Center app for Windows Phone, which caused issues for those who desired to utilise the app after upgrading their account to be more secure. Microsoft has now reported that it has fixed the problem.
So how does one get round the problem with the app not working? You can now pop in to the Microsoft Account management area to create an "app password" and use it on the Dev Center app. This will enable you to login and continue using the service as usual. If you haven't already checked out the Dev Center app, you can download it from the Windows Phone Store.
Source: MSDN; thanks, glass, for the heads up!