Basically(from what I can tell, and this may be wrong), the database server is occasionally being overloaded with connections, thus causing it to reject new connections, when this happens you get a database error on the forums(there are other reasons for a database error, but this is what appears to be the case)
The shoutbox is being blamed because with the nature of the software it opens a lot of database connections, due to this it aggravates the problem.
There are only a few solutions to this problem, either fix the SB so that it properly closes the connections(assuming that it isn't, I mean might be, idk), edit a config file that they may or may not have access to, depending on the host(which they could always upgrade to a host that allowed it), or just simply ditch the SB and call it good.
Now, that whole post was based off of the limited knowledge that I have, and there are probably details that I am not privy to that would change the situation, but that is my outlook.
So, tl:dr, trust the guys in charge(aka Mases and Locke), surely they can figure it out.