I'd like to bump this and get some more thoughts on it. I'm not suggesting this to be xenforo core though, I know this would be an add-on, just thought I'd bump it.
The 'like' feature is a great way to identify good/interesting posts and it seems a shame not to take advantage of this to deliver the best content to users. I works very well on sites like reddit.
Of course as
@Brogan says, it may jumble the conversation so it doesn't make sense. But if you look at most threads, people are simply responding to the original post, so it's not necessary to order by time. In these situations, I think it would work very well to then order by 'likes' to put the best voted posts at the top, rather than having to read the whole convo to get the best content.
The only time I can see this being an issue is when people are quoting other posts - this becomes an issue as it is time sensitive. To solve this, if you used the comment feature (or similar) https://xenforo.com/community/resources/*******-post-comments.2434/ then all quotes are simply added underneath the post, ordered by time posted (not 'likes').
I've been going through popular threads on my forum and re-ordering it in my head using this system and it seems to work very well and all makes sense.
In fact, it seems to be better than the current system as all the best posts are at the top and all the quotes are grouped directly under posts which keeps all related posts together. It also stops threads being derailed.
It's like the perfect blend between reddit and xenforo