Sometimes even the most inanimate parts of a webpage are constructed using Flash because only Flash offers the functionality the developers needed. For example, it’s common for web developers to implement file uploading capabilities as Flash elements because good ‘ol HTML uploading doesn’t give you any way to measure the upload’s progress. If you’ve ever attached a file in gmail, you’ve used a Flash uploader.
As to your original question, I’m not sure precisely what that Flash object in Facebook could be, but it’s a good bet it’s the chat feature like @downtide said.