FAF stores its patch files, movies etc in C:\ProgramData\FAForever\ and the installer doesn't give you the option to choose a different directory for that. If you use windows you have the option of using a NTFS junction to save space on your C drive.
C:\ProgramData\ and put it on a drive with more space. As example we'll use D:\(windows + R -> cmd -> enter)mklink /j C:\ProgramData\FAForever D:\FAForever to create a junction