Depending on the state, it can be the responsibility of the couple or the officiant to return the completed marriage license to its issuance office.
In most States the marriage license can be returned by mail, making it a moot point on whose responsibility it is to return the marriage license. In some states however, the marriage license needs to be returned in person by the couple or the officiant.
Specific instructions should be included with the marriage license when it is filed for by the couple. By the time you are ready to return the marriage license you should know how it is to be returned. If you have any doubts, don't hesitate to contact the office that issued the license and ask them.