A standard path for Firefox is /usr/lib/mozilla-firefox/
It shouldn't be chown'ed (of course you should have permissions for reading and executing), but you will not be able to update it thru FF's menu (Help->Check_for_Updates...) if you do not have write permissions.
mv /home/user/DOWNLOADS/firefox /usr/lib/mozilla-firefox
should help you, but I have no idea why it doesn't work from the homepage, may be
chown user -R /home/user/DOWNLOADS/firefox
is what you need?