Research has looked at chlorophyll as an antioxidant, and diets high in antioxidants might provide some protective benefit against certain cancers. Our bodies don't use chlorophyll in the fat digestion process, and I haven't found any scientific literature suggesting that it might "support" fat breakdown in the liver. Here's a summary of what evidence exists for Chlorophyll:
http://www.wellness.com/reference/he...al-information
With that said, even if it was a magical weight loss aid, there's no need to drink it in liquid form. Green veggies have a ton 'of chlorophyll, and if you don't overcook them, they'll give you THAT as well as a ton of other fiber and nutrients.