I've seen multiple reports on Google from people trying to login to Flickr through iOS 7 Settings and receiving the 'Could not communicate with server' login failure. I changed phones last week and received the same error message this morning. The fix that worked for me is:
1. Go to Flickr through a desktop browser.
2. Go to the your profile image in the top right and click Settings.
3. In Your Account select the Sharing & Extending tab.
4. Revoke the iOS related Account Links.
5. Login again on your iPhone through Settings and you should now have access.
I hope this helps.
Usual disclaimer: please remember that this is offered purely as information about what worked for me. I do not accept any responsibility for issues that may be caused by anyone else following these steps and you do so at your own risk.