Go to "tools"/ "options"/ "Privacy" /
Set option in box: "Upon successful download".
All recorded downloads will now be erased from the DL manager upon completion.
PS - it just deletes the listing, not the downloaded files.
So beware new FF users, this default is "Manually" and if you don't change it or empty it regularly - manually - eventually it will get so full that it will hang your system something fierce.