Learn English – A word or phrase to describe documents that can expire

I would simply use the term non-expiring. For example, in Washington state, one can obtain a non-expiring license for child care.

You may also encounter the phrase valid in perpetuity, e.g. Hawaii Rev. Stat. §481B-13 (about electronic gift cards/certificates):

…if the gift certificate does not have an expiration date, it shall be valid in perpetuity.

This phrase is generally used in describing the length of effect of a legal document (like a contract or a constitution) or financial instrument rather than an identification, though, so I would consider it a secondary choice.

In the (most) remote future?

"remote" - far away in time : happening a long time ago or far into the future. Merriam-Webster