The noarchive tag will prevent Google from showing the Cached link for a page in their search results. But it does not prevent the page from ranking well in Google says John Mueller.
John Mueller of Google replied to the question on Twitter saying “no” using the noarchive tag on pages does not negatively affect Google rankings for those pages.
— John ☆.o(≧▽≦)o.☆ (@JohnMu) August 29, 2017
I guess it is a logical question, whenever you restrict Google from doing something, maybe, just maybe it can have a negative impact. But no, not in the case of using the noarchive tag.
So feel free to use it if you don’t want people to see your cached page in Google.
Forum discussion at Twitter.