What's the difference between"next night" and " tomorrow night"?
And can we say yesterday night?

Question

Answer

When you are talking from the perspective of today you should use 'tomorrow night' and 'last night'.

'Yesterday night' is not so common

"Tomorrow night I'm going to study English"
"Last night I met a friend for dinner"

When you are telling a story often you might want to talk from the perspective of another day. In this situation we could use 'next night' of 'the night after'

"Last monday I went out until 2am, the next night I went to bed at 8pm because I was really tired"

Here 'the next night' is used from the perspective of the Monday already mentioned.

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I have never heard of yesterday night. I would say last night.

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As I imagine tomorow night means only  tomorow. Next night can be folowing night from point of view of any day. 
Yesterday night in my oppinion is ok.

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There is no such expression as "next night".

"Yesterday night" is acceptable.

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next night sounds super funny :P