I have a field whose value is constantly changing (I think it's called a burner field?). The field value is a customized URL that is assigned to any targeted list of leads, so that each lead has its own unique URL.
If I send out an email using this field (as a token), then send out a second email using the same field but with an updated field value (i.e. different URL) to same list of leads, will the first email's URL value still function properly?
If I am understanding your question correctly, then yes the link in the first email will work correctly.
Tokens are not dynamically linked to their fields or values. They are merged at send and the actual value at time of processing is sent to the recipient. Changes to the field value will affect future sends but not past ones.