To pretty much everyone who reads this site, this is a no brainer, but there are some people out there that think having a site with some 404ed pages will lead to a Google Penalty.
Sound crazy? Well, it isn't worse than thinking you need to noindex your site when you go on vacation. But that is a different story. :)
Yes, when a page on your site goes 404 (page not found), Google will eventually delist it from it's index. But a 404 means the page is not found. So Google won't list it because you specifically said there is no page. That is not a penalty, that is a directive you are communicating to Google and your users.
Gary Illyes from Google made it crystal clear, if it wasn't clear for you before:
Whoever came up with the idea that having 404s gives a site any sort of penalty, you're wrong. Utterly wrong.
— Gary Illyes (@methode) August 6, 2015
Then he was asked the question a different way:
@WLau89 Nope, no negative effect.
— Gary Illyes (@methode) August 6, 2015
Also, soft 404s do not lead to penalties, Google told us that also. But you can use 404s to help with existing penalties. You can 404 pages with bad links pointing to it, it kinda acts like a powerful disavow and bad links to 404 pages do not hurt you or the overall site.
In any event, 404s do not lead to penalties, but it does lead Google to deindex the page with the 404.
Forum discussion at Twitter.