Is an eye doctor one of the specialists a patient sees if he has a brain tumor?

Sometimes: Tumors in certain areas of the brain may become evident on an eye exam.

Yes.: If the tumor is of sufficient size to cause an increase in the intracranial pressure (icp), there is swelling of the optic nerves as they are "pushed" on by the high pressure. The result is a finding called "papilledema" which is a tell tale sign of increased icp. So, yes, for larger tumors the answer would be yes. Small tumors may have no impact, especially intiially.