While in 21st century English, "remember" and "forget" describe largely mental activity, in the old Testament they described actions as well as thoughts. To remember someone was to take action with regard to that person while to forget someone was to forsake him. Read, for instance, Hosea 9:9 or Ezekiel 33: 15-16.
When God promises to "remember not" our sin, He doesn't forget the sin (or forsake us), he promises that He will not act upon it in judgment of us.
Whatever blows your skirt up.
Bye.