However, the premise that this only happens because the startup in question was "free" doesn't hold weight. Call me a cynic but the recommendation to use "paid-for" services only to avoid this outcome just seems a little too self-serving to paid-for-only service PinBoard (as excellent as it is).
I don't have an exhaustive list of startup acquisitions that were then shut down, but here's a few just from a Google acquisition list I found:
* PostRank was a paid-for blog data service Google acquired, shut down
* Jambool was a virtual gold payment processing system Google acquired, shut down
* Gizmo5 was a VOIP service Google acquired and shut down.
In some cases even if a company is taking money that alone still doesn't make it viable. In other cases, like PostRank and Jambool where the $$$ being paid by customers was probably significant, acquirers like Google have other circumstances where they'll shut a profitable businesses for a 'higher goal' in a bigger project.