I think a VPS is a bit much for a little forum script. I guarantee that you can find hosting for less than $1/mo that you would be very happy with... unless and until your forum really takes off -- then you may need to move up the food chain, but not necessarily to a VPS.
A standard shared hosting plan would be fine, if you get large amounts of traffic at a later date then you could look towards a VPS or dedicated server at that point. A good host will allow you to upgrade your hosting as your website grows.
This would depend on then host and what you get in a shared plan.
Personally i would not run a forum on a shared plan as these could cause bandwidth issues due to traffic which would affect all others on the shared server and could lead to the host kicking you off for server abuse/fair usage abuse.
Shared hosting is OK for any today, modern software like xenforo, phpbb and similar, you can easily upgrade hosting with almost any provider then to VPS etc.