I will deliberately take a contrary position just for the sake of writing incredibly long arguments
Hypothesis: the closeness of a forum community increases as it approaches its death.
Here's a different take: I frequent this Czech forum belonging to a gaming magazine for ~15 years...
That's what I'm getting from this thread, anyway, and it seems pretty easy to bare out with examples. Let's start at the other extreme. I've gotten hooked on Reddit, and I've posted a grand total of about five comments. I usually browse with "hot" topic sorting, which means I don't even see a topic until it's proven to be popular. Then, I browse each topic's comments with "best" ranking, which boils every dialogue down into the few most popular comment chains. I assume that most Redditors do the same. This makes it feel pointless to join in any discussions at all - I'm always a few thousand comments too late to say anything new, and even if I had some brilliant revelation nobody would ever see it because the swarm of upvotes has already come and gone. This is the last place that you could ever connect with another poster on a personal level. Content is decontextualized, commoditized. I find it amazing that anyone in that place can even recognize anybody else's username, and then people are known more for their quirks than their personalities. There's /u/Unidan, the exciteable biologist, /u/Shitty_Watercolour who paints decent topical watercolours and... that's about it.
Old-school forums like Doomworld strike a reasonable balance. Without any built-in comment ranking or threading, you have to at least scroll past everything that's been said so far. Avatars give the primitive monkey brain something to anchor onto - you recognize the same posters from thread to thread and stitch together a sense of each one's personality without realizing it. The problem is that with so many people coming and going there's still a lot of white noise disrupting the signal. It's hard to get to know the old guard when their posts are spread thin by an influx of members with much more to say (because they haven't gotten sick of all the familiar discussions yet).
For my example of the extreme closeness of dying communities, I'm just going to point towards A Life Well Wasted, episode 2. There's a surprisingly moving documentary bit about the world ending, not with a bang, but with a "server disconnect" message.
As for personal experience, I've never really felt close personal connections on this forum or others. That's partly because of past experience; I used to be hooked on Runescape, and the characteristic relationship between MMO buddies is all work and no reward. The neediness of someone who has no real life and needs your devoted attention to justify their continued slavery to oppressive game mechanics is truly spectacular. I got soured on the whole "online friends" thing. Or, maybe I'm just anti-social, verging on solipsism, almost; I sent someone this in a PM a while ago:
My flippant responses are because I've never had much interest in trying to build a close and casual relationship with anyone that I'll never meet face-to-face. Despite my post count and registration date, I've logged into Doom-themed IRC channels a total of about five times and I've never had a private IM chat with anybody here. Posting for me is a personal, private game; my goal is always to amuse myself.