I’ve never actually used the Firefox download window for anything useful. I don’t really care about keeping a list of downloads, and anytime I accidentally open the downloads window, it seems to hang the browser for a few seconds.
Turning off the tracking of downloads is really very simple. I didn’t realize how simple it was until
Click Tools \ Options and then the Privacy tab.
Uncheck the box for “Remember what I’ve downloaded”. You might also want to clear the download list while you are at it.