First of all, the original question: Although I've never used Anyboard myself, it looks like it is a Perl script with local database files. It doesn't look like there will be any issues with installing it on Dreamhost, although I'll mention one possible caveat: YaBB is another similar piece of bulletin board software; it is written in Perl and uses a flat-file database on the webserver. Apparently, this database and/or the way it uses it is very inefficent, causing the software to bog down the server it is installed on. Unless you've got your own server, this is bad for everybody else on your shared server, so Dreamhost effectively outlawed YaBB to protect other people on the server. (I happened to be one of those people, and I don't have a problem with this--I upgraded to a much more efficient PHP/MySQL version of YaBB and haven't looked back.)
Whether Anyboard is more efficiently written than YaBB or not, I can't say (though my guess would be yes). If it isn't well written, however, and your forums see a lot of traffic, there might be an issue with the script. On the other hand, if it turns out there is, there are many good MySQL database-based forum solutions in existance, such as this one (PHPBB... I think?), and regardless of what Anyboard's marketing-speak says, those tend to be much faster than flat-file forums.
Now off topic, with everybody else: I've been with Dreamhost for 4 years, and I'm a happy customer. It's obvious that many people (or at least many that post here) have been having some significant issues lately, probably related to growing pains at Dreamhost. Hopefully this will be a temporary blip, but I don't know.
As for my personal experience, I remember a period about 3 years ago where Dreamhost went through a rough patch (also due to growing pains) with an unusual amount of outages. It got much, much better after that. I personally have only once been affected by any of the issues of late (the SQL servers going down for a few hours in the middle of the night during a datacenter move).
I'm not saying that there aren't problems, and I hate to see other customers having obviously serious issues with their hosting accounts, but I haven't seen them personally, so it's not universal. In fact, I believe that things are better than they were when I signed up, and I picked Dreamhost due to their high quality at that time.
I may be proven wrong in the future, but I have some faith in Dreamhost getting things smoothed out. The issue isn't one-sided, so you should draw your own conclusions.