If my lease has the wrong dates on it, can my apartment manager make me stay 3 extra months?

When there is an error or ambiguity in a contract, the courts, in the event of litigation, will try to determine what the parties' actual intention was and enforce that. You could move out, and if the landlord tries to sue you for the rent you allegedly owe, you would defend yourself in court by pointing to the 12-month length of the lease and attempting to show that the error was in the specific dates (e.g. move in date) on the lease--i.e., you'd try to show that the 12-month duration was correct.

However, be aware that it is not a given you would win; another plausible interpretation, consistent with the facts, is that you signed a lease for 12 months, from, say, May 1, 2011 to April 30, 2012; but that you also rented on a month to month basis for three months prior to the May 1 start date. In that case, you'd owe a total of 15 months rent and could be held or obligated to the end of the lease period.

If there is "extrinsic" evidence supporting your case--for example, emails or other corresondence showing that you were only ever intending to rent for 12 months total, that would clearly help you; and conversely, any documents showing that you intended originally to rent for the 12 months listed in the lease, but then need someplace to stay for the three prior months and the landlord accomodated you, would hurt your case.

Since winning is not a given, and litigation has its own costs, you may wish to see if you can settle with the landlord--maybe pay 1 to 2 months rent. If you can and do settle, get the settlement in writing.