I was giving this some thought and my initial reaction was to suggest a Gmail account with +aliases, e.g. mythrowaway@gmail.com, and then use mythrowaway+whateveralias@gmail.com combined with a forwarding filter.
I then saw the 'own domain' requirement leading me to think Google Apps will let you do this (for money), and you could use a catch-all alias rather than having to use +subaddressing.
That said, if you want to use your own domain and host it yourself, you could easily do this on something as small as a Raspberry Pi with exim/postfix, and just update /etc/aliases as required.
example /etc/aliases file:
throwaway1: realaddyaliassecret
throwaway2: realaddyaliassecret
throwaway3: realaddyaliassecret
throwaway4: realaddyaliassecret
realaddyaliassecret: myrealaddress@yahoo.com
Note that you don't need to use a 'realaddyaliassecret', it just makes life a bit easier than having to always type in your real email address. And when you're done with a throwaway, just remove the entry from your aliases file, and messages should now bounce.
Shouldn't be particularly difficult to build a nicer front end too, cli or webbased.
EDIT: all of this now has me wondering why I haven't done this for myself already, given how simple it is. Thanks for the idea!