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
(windows + R -> cmd -> enter)
mklink /j C:\ProgramData\FAForever D:\FAForeverto create a junction