Here are the general rules for getting private school tuition reimbursement
1) The District MUST offer Free Appropriate Public Education (FAPE) in the Least Restrictive Environment (LRE) to children protected under the IDEA
Appropriate means reasonably calculated to allow the child to make meaningful progress based on the information everyone had at the time the IEP was developed.
Expert testimony is necessary to support your argument that a FAPE was denied. This generally comes from evaluations of your child. You have rights in this regard:
The District must evaluate children for all suspected disabilities. The District must reevaluate children at least every 3 years. Parents are entitled to disagree with the evaluations and, in writing, ask for independent evaluations. Note that the District can respond by bringning a Due Process Complaint to defend its evaluations
2) If you feel you have been denied a FAPE, based on information that you shared with the District, you can place your child in an APPROPRIATE private school. Appropriate here means that the private school providing specialized instruction meeting your child's unique needs. Here, again, you need expert testimony to support your argument
3) The parent must cooperate with the development of an IEP and consider all options offered by the District. The parent must share all private evaluations (those they intend to rely on in the future dispute). The parent must provide 10 day written notice that they are withdrawing the child from public education and why they are doing this. Tuition reimbursement can be reduced if the 10 day notice isn't provided.
At that point, a due process claim is filed. This is followed by a 30 day negotiation period called a "resolution period." Otherwise, claims are heard by Impartial Hearing Officers (IHOs) in a due process hearing at which evidence and testimony are presented. In New York, IHO decisions may be appealed to a State Review Officer (SRO). Generally, no new evidence is considered on appeal. SRO decisions may be appealed to State or Federal Court.