Mass Effect 3 does not only have no partial install on the PS3, it does not have any install at all (unless I severely fell asleep when starting the game).
The actual space required, like with many BioWare games, depends on the size of the save game files, which in turn depend on how far you are through the game. They won't reach the mammoth proportions of the Bethesda games however, so you should not have to worry.
As general advice however, you should keep a few gigabytes free on your HDD at all times to allow the system some space to swap in and out virtual memory. You might also want to consider upgrading the HDD or replacing your system. (I did this recently because of the problem in this question ).