Most of the activity on forums, is at the database end, not the web server.
However many people start out by setting up a low end dedicated server with both the web service and the database server on it, and find it's quickly overwhelmed.
Some forum code is poorly written - for instance I've seen off the shelf forums which implement the Search function by running a WHERE PostContents LIKE '%SeachTerms%' query which is an absolute database killer.
If you have virtual hosting now, you may not be able to see all of the necessary statistics about what is draining the performance. A danger with virtual hosting is that eventually, at some point, the host may tell you to go elsewhere because you are "monopolising resources" - some hosts, who I won't name, just shut your account without warning.
Worth bearing in mind that performance problems can be caused by over-utilisation of shared servers at the host, but if you have 1000 users online at once, this does point to it being very busy.
You don't say what the code is or what platform it operates on, but I'd have thought the option which would give the most for the money would be a small dedicated server box for the database server (MySQL or MSSQL) and possibly a virtual or cloud hosting account for the website.