This just started for me on Firefox as well, maybe in the past 2-4 days (give or take). The only way I found to stop it was to change my browser settings to always ask me for the location to store downloads. Then when the prompt comes up I click Cancel to stop the download.
The download files have an extension of .htm and the size is 0. Not sure what these files are, but I only get the prompt here on DISboards. It doesn't appear to be isolated to any particular forum, which makes me wonder if it's somehow ad-related. Due to past issues with ads here, I use AdBlock now so I don't see what ads are trying to display.