Hello jordisan10,
Thanks for using the Apple Support Communities. It sounds as though you're having an issue with Contacts on your iPhone. You mentioned that the birthday information goes missing, shortly after it's been added.
Have you noticed if other information, such as an address or note, when newly added, goes missing as well?
Are you storing your contacts in iCloud or is that information linked to a different account, such a Gmail or Yahoo? If you're not sure, you can check this in Settings > Passwords & Accounts > Accounts.
We noticed that you listed iOS 13.6 as the iOS version currently installed on your iPhone. Recently iOS 13.6.1 was released, so updating your software is a good troubleshooting action to perform to help with this issue.
Update your iPhone, iPad, or iPod touch
Best wishes.