If you are talking about having a ‘fear of something’ you always need to say ‘a’. If you are talking more generally, either ‘a’ ‘the’ or no article can sometimes be used.
Dad developed a fear of heights. (Article required, not ‘the’.)
The/a// fear of open spaces is called ‘agoraphobia’. (All 3 options work but I would naturally say ‘the’)
Q. What’s your biggest fear?
A. The fear of snakes. (Article required, not ‘a’. You are already talking about fears. This specifies which one)
Q.Why didn’t he take the job.
A. Fear of the unknown. (The answer wasn’t necessarily a fear)