Forums use a lot of RAM since they tend to keep the tables for the database in memory. Depending on the size of the forum, amount of posts, and number of users, you could get by on a cheap hosting package or need a VPS. Generally though, you'd be fine on a cheap shared hosting plan.
I'd recommend you to go with a host that allows you to grow without having to move everything all the time. I started out with Siteground, and that was fine until I actually needed support. I eventually got tired of it and moved to Verio. They have some very nice offerings for plans as you go and are actually there when I need them at 3 AM in the morning if something goes crazy.
You can do a lot of stuff on shared hosting, and even extend the life of the plan by changing how the database caches site data, etc. So don't spend a lot of money until you have too.