Anyone reading that uses Twitter - would you pay or tweet to use a service? I'm guessing the vast majority would tweet... but am really not sure :)
In general forcing viral activity both misses the point and fails to work. I'd say you should at most randomly give away free inbox cleanings to your followers / people who tweet about the service. Really, though, you should probably just be active on twitter and talk to people. That doesn't scale well, but it's the most effective viral tactic there is.
This works because you're offering something that normally costs money for free. That's psychologically different than offering something for free (with a required plug), but with no option to pay for it.
The other thing is that it needs to be an impulse purchase. I doubt it would work for a Shoppify, for example.
So if you're selling something sort of whimsical, entertaining or trivial, this sort of "forced viral" marketing can work very well.