How hard would it be to automate a cron job to review all the tasks in
the database every 10 minutes and send out an email?
Do you really need to be looking for the best FOSS task management
software and then consider how to mod it to your requirement?
I've never used a web based solution in anger, but I thought this was an
interesting thread, not sure why you think it's OT.
If you find something FOSS you like and would like some help adding
automated emails, then yell out, not sure how much I can offer, but I'd
like to help.
Cheers Don