Proper "notice period" for resigning a tenure track position?

Hiring an adjunct or borrowing someone from another department for one course is about the least bad thing that can happen in this world. Giving the kind of notice (9 months?) that would have allowed them to do a full candidate search is impossible for folks who move laterally out of a department. This is pretty common and departments simply have to manage it. Everyone knows that offers are being locked in during the April-May time frame, and sometimes you lose someone key. It would not be unprofessional if you chose to avoid delaying your career by staying for an extra semester in order to make your current department's process painless.

Nobody wants to see you go, but you're working within the system to the best of your abilities. You're not screwing them, and it's not unprofessional to wait to give notice until the new offer is accepted.

The American Association of University Professors has established ethical guidelines for hiring faculty from one institution to another. Perhaps the most relevant clause in their 1993 policy document "The Ethics of Faculty Recruitment and Appointment" is the following:

An offer of appointment to a faculty member serving at another institution should be made no later than May 1, consistent with the faculty member’s obligation to resign, in order to accept other employment, no later than May 15. It is recognized that, in special cases, it might be appropriate to make an offer after May 1, but in such cases there should be an agreement by all concerned parties.

The American Association of Universities adopts a slightly stronger stance:

We believe that a responsible approach for both institutions and the faculty members would be to consider offers made or pending on May 1, or thereafter, to be effective normally only after the intervention of an academic year.

In my experience, US universities that follow these policies do so as follows. Suppose university X wants to hire someone on the faculty of University Y, to start the following fall (typically August 15). If the offer is made after May 1, the appropriate dean at University X first officially requests permission from the appropriate dean at University Y to extend an offer. The dean at University Y will usually agree, especially if the request is made early in the summer, but for late requests or other special circumstances, the dean at University Y may request (or "demand") a deferral.

There is no legal force to these policies, but following them is widely considered good professional practice — bluntly, the protocol is enforced by peer pressure. Deans that don't agree to reasonable hiring requests may find their own faculty hired away without their agreement later. Also, these agreements bind institutional behavior, not the behavior of individual faculty; under normal circumstances, you do not need your dean's permission to resign. (There are some exceptions; for example, my university requires faculty on sabbatical leave to either return to campus for one year after the sabbatical ends or repay their sabbatical salary.)

So if you were in the US, your current dean, and therefore your department chair, would already be aware of your offer. Even if they don't, the AAUP guidelines strongly suggest that three months notice (May 15 to August 15) meets any professional obligation to your current university.