BTW, I assume you want to have a good experience here at DreamHost and so you want to make darned sure that your site is appropriate for the kind of shared hosting that DreamHost can support, right?
I would contact DreamHost support via the link below, tell then which forum you're running and give them an idea of what your load is from a posts-per-day, pageviews-per-day, and searches-per-day perspective and ask them if they can support that kind of load on a shared hosting plan. You might even give them the url for your current forum to help them.
The fact that another shared host is having a problem with hosting you (albeit due to a fixed limit) is a yellow flag that means you should take care not to take the answers here as meaning that you have unlimited usage of a database server. There is a limit, it's just not fixed to the number of queries you run in any one hour period.
Here's the contact url:
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