Xcode build warning - Target Integrity - Provisioning profile is expiring only for the ProjectnameTests Target
I had the same issue.
Worked out in the end it was Xcode warning about some expired profiles generally - not ones specifically linked to the current project.
To fix I went to Organizer/ Devices Tab and under Library select Provisioning Profiles
I had two profiles in the list that had expired - I didn't need them so I just right clicked and deleted them.
Warning then went away.
This is a bit different with Xcode 5 now when working with Teams ...
I go the same warning of an expiring old profile which didn't exist anymore - but it still did exist in the local Account setting for the connected team in Xcode (Xcode -> Preferences -> Accounts -> <team account in question> -> View Details). So I just did a refresh on these profiles, and yes, the warning was gone.
Unfortunately there are many more expired profiles for other apps in this team account which I just synched into my local Account setting (and I can't just remove them, as it's a client's account ..) - and now Xcode complains about all of them being expired. Guess you should be happy with what you got sometimes, it can get worse ...