In order to log-in using FB on the iPhone, you'll probably have to be logged-in to FB on that phone (using the FB app), and then start the Chess app and press the log-in using FB option. The Logging-in using FB option probably won't work If you're not logged-in on FB on the phone.
Can no longer log in from Facebook

just did a google search. from pspizzo on forums, seemed to have worked with a lot of other members having the same problem after an iOS update in 2015: Go to your iPhone settings and find the "Facebook" section. In those settings, enter your email and password to log into Facebook (once I logged in, I turned off Calendar and Contacts syncing). Now when you launch Chess.com, it will ask if you want to allow it to access Facebook. Approve the request and you'll get in.

I clicked “Find freinds” in the new game ->custom game section and then i got a message from fb that my fb account is locked, what happened ?? Please someone help! Im still young so i dont have any gov issued IDs so cant unlock it...

I clicked “Find freinds” in the new game ->custom game section and then i got a message from fb that my fb account is locked, what happened ?? Please someone help! Im still young so i dont have any gov issued IDs so cant unlock it...
You'll have to contact Facebook about that. Or, if your email is attached to and old Facebook account you may just need to unlink your chess.com account first.
https://support.chess.com/article/267-how-do-i-change-my-google-or-facebook-account
I changed my Facebook password and I can no longer log in using Facebook from an iPhone 6. I uninstalled the app and reinstalled, didn't help. Any other suggestions? Thanks!