Will fat transfer cause your face to weigh down?

The short answer to your question is no.  But yours is a great question and one that has a lot of myths and misconceptions surrounding it.  
According to elementary physics, since fillers and fat both have mass (aka weight), placing mass in into anything will "weigh it down" not "lift it up!"  Alot of promoters of fillers talk about the Liquid Facelift.  They criticize surgical facelifts (fake, dangerous, etc.) and imply or claim that filling the cheek, for e.g., lifts the jowl.  You will notice that ZERO scientific studies support this claim.  Put an ounce of filler in your right cheek and an ounce of pennies under you left, and what will occur?  The filler will lift and the pennies will weigh down?  That is not even logical.  But even though replacing volume does not "lift", it can, (if placed artfully) rejuvenate the area and make it appear more youthful and attractive.
As far as fillers versus fat.  Both fat and hyaluronic fillers are of very comparable density. Bear in mind that most fillers are provided as only 1cc (only one fifth of a teaspoon) at a time.  So typically only a few cc's are added. In fat grafting, a wide range of much larger volumes- from 20-60cc or more- can be injected.  But even in my largest fat grating cases, I have not seen even a hint of weighing down of the face.  But I have not seen appreciable lifting either.  I guess that while the fat does indeed add mass (or weight) it perhaps serves a subtle supportive function as well.  A baby's cheeks are very full-  but they do not droop. 

Thank you for your question. No, However swelling is typical after fat transfer and can last 3-6 months. I recommend prp to help improve final results. Icing the area, predisone, ibuoprofen and venus legacy treatments can help reduce swelling. See an expert. Best, Dr. Emer

well it depends how much you put in who you do it on and if the fat grows or not. As peoples faces get heavier just with age this could be considered "weighing down" , though it's really just an expansion which changes the shape of the face.    YOu can get that from too much fat or filler. If the fat grows it can turn a nice result into a heavy looking face

The answer is no.  Filler nor fat will not weigh your face down.  Consult with a highly experienced board certified plastic surgeon for an in person evaluation.