Firstly, I think Webinsane nailed it -- chat should only be for closed groups where anyone can create their own chat 'room' and invite whowever they want to join (maybe set a limit on the # of participants), but the chat window/content is not visible to anyone else. Everything else belongs on the forum else you risk chat taking over as the main way members communicate.
Secondly, you better have a solid community before you think of adding chat.
Funny story here; you might not think it's related to chat, but it really is because we are essentially talking about giving users a quick and convenient means of communication.
Anyway, on with the story...
...One of the off-road dirt bike forums I belong to uses vB5, and most members simply hate it. So, I began testing alternative forum software: Flarum, NodeBB, and Discourse.
To keep our 'new forum software' discussions private, I setup a small Slack channel and invited a handful of members. Please keep in mind that none of the participants have any background or even any interest in IT -- they just want their new form software to be easy to use on their phones (i.e., ability to easily and quickly post discussion and insert photos of their rides from their phones). None of them had used Slack before this.
After using Slack for only 2 days, NO ONE wanted to use a 'traditional' forum anymore, including Flarum, NodeBB, or Discourse. They all wanted something as easy to use as Slack! When I asked why, the answer was unanimous -- no web browser. Everyone loved the fact that they did not have to fire up a web browser and navigate to the site and login, etc., to start communicating.
- If they came upon a downed tree across a trail they could make it known to the rest of the community (channel) within seconds using Slack.
- If one of their riding buddies just had a spectacular crash they could snap a pic on their camera phone and it would be up on our Slack channel in seconds.
They also loved the fact that Slack had a very convenient and easy to use desktop app.
Of course, Slack can not (currently) replace a forum because it lacks the ability to have a structured 'forum-like' discussion. Secondly, Slack is not free (in any real world use-able state).
I have to be honest here; if Slack was free and included a 'Posts' feature I would not be here at all. I emailed the team at Slack about adding these features and also offering a self-hosted version and was told they were already evaluating some of those options.
So what's the moral of the story?
Be careful what you ask for! If your forum is not 'solid' (content-wise), adding chat might just derail your forum altogether.
Sorry for the long post!