Varnish can only really help Reddit with users who aren't logged in since every page is customized. It'd still help in high volume scenarios since Varnish solves the thundering herd problem: only one request (at most) will hit the database for each invalidation.
Vanish can cache snippets of content. They refer to this as Edge Side Includes.
Reddit's customization is far more than snippets. I've used ESIs with Varnish extensively, and I don't think it would help much for authenticated Reddit users.
Also, if you were to use Varnish as a cache, that wouldn't help you much with write access because every time there is a write (a new post..) you'd have to invalidate the cache. In a high volume scenario such as this (tons of posts coming in..), the page is as good as dynamic, even if you DO cache it for a bit.