Hi,
I'm with Philip on doing this selectively. Many users find content hard to digest in the manner it was intended when they have no idea if it was written in 2006 or 2014. If your industry is one where time really doesn't matter or change anything, this may be different. From an SEO point of view, Google still knows when it first crawled a piece of content, so not including a date won't fool Google into believing the content is fresh when it's not.
I agree that creating evergreen content where appropriate is also a good move - articles, case studies, etc. that can be updated where appropriate (and blogged about + linked to to spur re-indexing).
If you see a marked drop off in traffic to or engagement in a post from 2+ years ago that used to be successful for the site, I'd consider re-writing the post with a current slant, linking to the old one from the post and being upfront about the fact that you're refreshing an old article. You can move engagement to the new post. If there really isn't anything new to say about the subject, you can still cover it again "for new readers" and redirect the old post to the new one, canonicalise it or leave it as is if there are no duplicate content issues.