There is a lot more clarity from Google these days around crawl budget which is why I found it a bit surprising to see some SEOs ask if using a nofollow attribute on your links can hurt your crawl budget. We even reported some time ago that using the nofollow in some cases can help with crawl budget.
But Google's John Mueller had to say that using nofollow does not hurt crawl budget. He said on Twitter that it is incorrect to say nofollow burns crawl budget, it does not he said. He added that anyone who says that "don't know what they're talking about."
Here are the tweets:
@_nitman @dawnieando If they say "nofollow burns crawl budget" they don't know what they're talking about. Ignore them :)
— John ☆.o(≧▽≦)o.☆ (@JohnMu) May 29, 2017
I am not sure where this myth came from to be honest.
Forum discussion at Twitter.