your current director could be a tremendous resource to tap for questions because she's aware of the needs and demands of the job as well as the existing staff, parents and children.
look to the duties you want the new director to handle and formulate questions based on those. i also think since a preschool director is responsible for writing memos, personnel reviews and for reviewing material her staff will provide parents that an ON THE SPOT demonstration of writing skills can be very valuable (people can produce incredible resumes, cover letters and the like with the help of computer programs and outside help, having a spontatious writing componant can detect issues/problems not otherwise apparant until it's sometimes too late).
if the director is responsible for supervision then i think questions exploring that person's understanding and knowledge of personnel practices is imperitive. a situational question that has implications of fmla or ada are important, knowledge of your state's employment laws-and specificialy laws governing lic. daycare and health and safety are vital. knowledge of requirements pertaining to mandatory reporting to social services and the issues of staff/parent/student confidentiality are essential.
finaly, if your school has a diverse student body then exploring if a candidate reccognizes the differences in the way some cultures interact with academic professionals is important. depending on the culture, the manner in which a teaching professional approaches a parent with a question or concern can be interpreted wholly different than the professional intends and can cause a total break down in the necessary flow of information.