Question:
 Can the Imam’s house be built above the masjid?

Answer:

In the Name of Allah, the Most Gracious, the Most Merciful.

As-salāmu ‘alaykum wa-rahmatullāhi wa-barakātuh.
 
It is very important that the trustees of a masjid consult the ulama with regards to the specifications of a masjid prior to establishing a masjid.  A masjid should only be established after thorough thinking and assessing all the outcomes.
 At the time of establishing a masjid, by default, the space parallel to the masjid above and below will be considered to be part of the masjid.  However, if a firm intention was expressed to exclude a certain section from the space parallel to the masjid, that section will be excluded. 

This has to be known before the masjid is finalised and established.  If the trustees claim to have such an intention after the masjid has been established without mentioning anything prior to the establishment, their claim will be not be accepted.  As a consequence, it will not be permissible thereafter to exclude any parallel space from the masjid.[1]  

If the Imam’s house was intended to be directly above the masjid at the time of establishing the masjid, then it is permissible to reserve the space for the apartment.  However, if such an intention was not made, it will not be permissible thereafter to exclude the space directly above the masjid for a living quarters.  

And Allah Ta’āla Knows Best