Is there a word for following a man or his ideas too closely?

That word is ape (implying imitate blindly).

"Its important to not ape another man or his ideas"

ODO:

imitate
VERB [WITH OBJECT]
Imitate (someone or something), especially in an absurd or unthinking way.

‘The guy's just aping his mentors, ill-equipped to blaze his own path.’
‘Incidentally, the kids are only aping the behaviour they witness in real life.’

In order to convey the concepts of 'following too closely' and 'obsessing' and 'being blind to independent thought' I would choose the word 'idolize'.

trans (broadly) to love or to admire to excess

Merriam Webster

to admire and respect someone very much, often too much

Cambridge