For an aircraft to be approved for commercial usage (like Boeing 747s or Airbus 320s), is there any minimum weight requirements, just the aircraft (without passengers and crew).
While there is a minimum weight to be defined as Transport Category, 12500 lbs for jets and 19 seats/19000 lbs for propeller driven airplanes, that's a "minimum maximum weight" you could say, to meet a category definition, which determines the certification standards you have to meet.
There is no minimum weight for certification in any category. You define the mission you are designing for, payload, range, speed etc., and design it to be as light as you can possibly make it and still meet certification requirements for structural strength and performance (while allowing fudge factors for mistakes and flaws), while meeting your design goals.
So if you could design a 747 with new miracle materials, so its empty weight is 300,000 lbs instead of 400,000 lbs, and it meets all the Transport Category requirements for strength and performance, and it can still do what you want it to do in terms of speed/payload/range, you're good to go.
If you are referring to a minimum weight requirement that the aircraft must be operated with after it has been certified (that must be submitted as part of the certification process), yes there is. For example, for my A330's Weight Variant, in the Airplane Flight Manual's Limitations chapter, while there is a Maximum Take Off Weight (MTOW) of 233,000 kg, there is also a specified Minimum Weight of 121,000 kg.