Is there a word for the (probably irrational) fear of learning something new and maybe forgetting something you already know?1
The word I suggest covers more territory than the narrow sense you mention. It doesn't touch directly on the sense of "and maybe forgetting something you already know", although that is a component of the fear commonly expressed by those who suffer from it. The word has been in use for 130 years, at least:
A tendency to dislike anything new; fear of novelty.
(Collins English Dictionary – Complete and Unabridged, 12th Edition 2014. S.v. "neophobia." Retrieved May 5 2016 from http://www.thefreedictionary.com/neophobia )
A very similar term, rarer in use than 'neophobia' but as longlived, is this borrowing from Italian:
Hatred or fear of change or innovation.
(American Heritage® Dictionary of the English Language, Fifth Edition. S.v. "misoneism." Retrieved May 5 2016 from http://www.thefreedictionary.com/misoneism )
For your narrower sense, but omitting "something new", which seems redundant when used to refer to things to be learned, a technical term from medicine works. As with the broader terms 'neophobia' and 'misoneism', the word for the specific phobia doesn't encompass the "and maybe forgetting something you already know" dimension:
Morbid fear of learning.
(Segen's Medical Dictionary. S.v. "sophophobia." Retrieved May 5 2016 from http://medical-dictionary.thefreedictionary.com/sophophobia )