There does not appear to be a scientific name to describe the fear of Facebook. Perhaps the fear stems from a fear of 'being too vulnerable' or 'too visible' to the rest of the world. It may be that Facebook is just not the 'thing' for you and you'd rather spend your time doing something else.
Unless your fear prevents you from meeting with close friends or intereferes with your happiness, then there is probably no need to worry. If this becomes the case and you feel too 'closed off' from the world, or find yourself 'closing off' from society, then please consult with a doctor to help you address your fear. Many people overcome various degrees of social phobia with appropriate treatment.
If you would like to make one up you could start by looking at the latin words that mean both face and book, visio and libri, then we would take it and combine it to visiolibri. Visiolibri meaning facebook we just tack on 'phobia' so we get Visiolibriphobia
Greek: Prosopo (face) + biblio (book) + phobia gives Prosopobibliophobia. Why mix Greek and Latin?
Your question asks what the phobia of a fear is (X-phobia-phobia): the PHOBIA OF THE FEAR of the number eight would be octophobia-phobia. The fear of the number eight would simply be octophobia.
A phobia is an irrational fear phobia phobic means the fear of fear itself
Phobia means fear.
If failing is your fear the name for the failing phobia (phobia mean fear) is atychiphobia.
Necrophobia Phobia name for the fear of death. It is a common phobia
the only phobia of the fear of crickets is crickephobia
Turophobia - fear/phobia of cheese.
perichoraphobia. perichora (suburbs) + phobia (fear)
Bebuphobia...that is the fear or phobia of blueberries.
To have a phobia is to have a fear.
Mud Phobia is the phobia (fear) of mud. Its called mysophobia, myso meaning mud and phobia meaning fear.
Phobia means "fear of." For example, arachnophobia is the fear of spiders.
Atychiphobia (kakorrhaphiophobia, fear of failure, fear of failing, failure phobia).
Theophobia- the irrational fear of a god or deity. Remember its only a phobia if its an "irrational" fear, a normal healthy fear is not a phobia.
We use the term phobophobia to name the fear of fear or of phobias. (The word phobia comes from the Greek for fear.)
Emetophobia is the name of the phobia associated with the fear of vomiting.
Phronemophobia is the name of the phobia relating to the fear of thinking
Phasmophobia is the name of the phobia relating to the fear of ghosts
Aurophobia is the name of the phobia relating to the fear of gold
Melophobia is the name of the phobia related to the fear of music.
Mycophobia is the name of the phobia related to the fear of mushrooms
Mottephobia is the name of the phobia related to the fear of moths.
Selenophobia is the name of the phobia relating to the fear of the moon
Ballistophobia is the name of the phobia related to the fear of missiles
Eisoptrophobia is the name of the phobia related to the fear of mirrors.