Why is it,
Even some of the winner’s supporters fear Egypt’s divisions will be far from healed.
and not these:
*Even some of the winner’s supporters fear Egypt’s divisions will be far from heal, or
*Even some of the winner’s supporters fear Egypt’s divisions will be far from be healed.
* denotes ungrammatical sentences
This is basically about participles--those verbs in -ing and -ed forms, which could be used like adjectives.
To provide you a little background, let's consider:
She is beautiful. -- This means she is beautiful.
She is far from beautiful. -- This means she is not beautiful.
This is obvious because beautiful is an adjective. Now, let's try to explain a girl who just came back from a visit to a town called Springfield.
She is missing Springfield. -- She is thinking of the town, and might want to be there again.
She is far from missing Springfield. -- This sounds like she doesn't like the town much.
What if we write miss instead of missing?
?She is miss Springfield! -- Read it aloud and it might sound like she just won a beauty contest!
?She is far from miss Springfield! -- Isn't she beautiful enough to win the contest?
?She will be miss Springfield! -- Same as ?She is miss Springfield, but in the future.
?She will be far from miss Springfield! -- Same as ?She will be miss Springfield, but in the future.
? denotes questionable sentences
Of course, those are just funny interpretations, but I think they show why we shouldn't write will be far from heal or will be far from be healed really well. As you appear to know, heal is a verb.
To choose between will be far from healed and will be far from healing (which is incorrect for your sentence), I'd like to suggest reading about "present participles" and "past participles" in your grammar books. To help you a little bit, here is my crash course for participles.
If you think it makes sense to rephrase your sentence in the active voice, use the present participle form, which is the -ing form.
If you think it makes sense to rephrase your sentence in the passive voice, use the past participle form, which is the -ed form.
For example,
Who's the fat man sitting in the corner?
Why do we use 'present participle' (the -ing form)?
Because that fat man "is sitting in the corner".
Most of the people invited to the party didn't turn up.
Why do we use 'past participle' (the -ed form)?
Because those people "were invited to the party".)
Now, back to your sentence,
Even some of the winner’s supporters fear Egypt’s divisions will be far from healed.
Why do we use 'past participle' (the -ed form)?
Because what those supporters fear is: Egypt’s divisions "will be far from healed."