It depends on the size of your forum. For a small forum, you should probably keep it on shared hosting unless you need a VPS or dedicated hosting for other purposes. A VPS or dedicated hosting for small, low-traffic forums is excessive and would be a waste of money in my opinion. When your forum gets much larger and starts getting more traffic, a VPS or dedicated server will be necessary. Forum Promotion, for example, saw the need to move to a VPS around the time that we reached the one million post mark. Your need for a VPS or dedicated server, however, will be based more on traffic than on the size of your forum.
Shared hosting is the best solution to host your forum. You can upgrade to VPS or dedicated once your website has many visitors.
The reason why I am suggesting to go for shared hosting is for starting few months you will not have many visitors. For a forum to grow it takes long time.
Hello, see it all depends on traffic that your forum would be receiving. If you are using shared hosting, and it is able to handle coming traffic to website, with allocated resources, no need to upgrade another service. But in case forum visits increases which exceeding your utilization of resources, you need to move on VPS or Dedicated servers.