A "proposed site" and a "beta site" are two different things, so I'll address both of them separately.
A Proposal (i.e. a proposed site in Area 51)
There is currently no time limit for proposal to find enough supporters to launch — although we are currently considering a time limit on the Commit phase (this has not been decided).
Proposals that take painfully long to find people to sign up inevitably run into problems in beta. Half the users don't show up and the enthusiasm has long since waned. We've tried to "age" commitments to account for the older supporters that have moved on, but what Area 51 needs to do is capture the momentum going into launch.
If the folks who defined the proposal (the Definition phase) aren't ready to sign up in those first few weeks (or at least the first few months), the site will not work. Anecdotally, we "rebooted" our Mathematica proposal and, after 17 months of lingering, it raced back to launch in a record-breaking 32 days the second time around, and created a fantastically-successful site.
There is no explicit time limit for a site to get through beta. As long as the site is plotting a steady course, it is allowed to stay in beta as long as it takes.
When Will My Site Graduate?
Does this site have a chance of succeeding?
Two issues that can lead to a site's closure: If the quality is not sufficient to make the subject better on the internet, or a continuing decline in traffic that renders the site itself inviable.