Q:

Which parts of speech are 'cause'?

In this question:

What problems will this cause for individuals and society?

Which parts of speech are cause in this question?
If it is a noun, why isn't there be before this cause?
If it is an adjective, what is the role of this ?

A:

'Cause' in the sentence given is neither noun nor adjective, it's verb.
To understand this try answering your question - This will cause the problems like .... to the individuals and to the society.
cause (noun) - the reason behind some action. 
Examle: The cause of our country's plight is corruption. 
Or as you asked use of be look at this:
Corruption is the cause of our country's plight.
to cause (verb) - to make something happen. 
Lack of exercise causes ill health.
https://dictionary.cambridge.org/dictionary/english/cause

