By default a multimember LLC is taxed as a partnership. My answer presumes this LLC is being taxed as a partnership.
There is not any specific role or person that has to file that Form 1065 for the partnership but for who should sign the return there are instructions at http://www.irs.gov/instructions/i1065/ch01.html#d0e690
Form 1065 is not considered to be a return unless it is signed by a general partner or LLC member manager. When a return is made for a partnership by a receiver, trustee or assignee, the fiduciary must sign the return, instead of the general partner or LLC member manager. Returns and forms signed by a receiver or trustee in bankruptcy on behalf of a partnership must be accompanied by a copy of the order or instructions of the court authorizing signing of the return or form."
The LLC member manager can sign and file the return, in general.
In some cases the operating agreement for the LLC may include other restrictions or instructions that the member manager would need to follow beyond and above the IRS instructions for filing. Absent such restrictions, the member manger can sign the return and file it.
Please ask if you need more discussion (or there are other facts not yet stated) or clarification.