Should I store my Facebook access tokens?

There are a number of explanations here:
https://developers.facebook.com/docs/facebook-login/access-tokens

The time it takes for your server to talk with facebook does contribute to a good amount of slowdown. So it depends on how intensely you plan to use the api.
In that case, storing the key and other fb related user data can cut that cross-talk considerably.

If you are just using facebook as a login authenticator I don’t think you need to bother.

Cheers.