I would look to the Event Viewer logs to see if there's anything there as well as the SharePoint logs, as Business Portal really doesn't have any logging of its own.
That being said, I can't say why Business Portal would go down and then when you reboot the server it starts working again. Definitely nothing I can say I've seen in regards to Business Portal, usually it either works or doesn't work.
By rebooting the server, most likely resources or connections are being dropped, which may be helping BP then work, so I would also look at the amount of CPU and RAM being used on the SharePoint/BP server as well as the SQL Server holding the Dynamics GP/SharePoint/BP databases, to see if anything there is pointing to resources being consumed at a high mark.
When you say BP is going down, what exactly is it doing or showing? When it does this, can you still access the top-level SharePoint site that BP is installed onto? (i.e. if access BP via http://servername:port#/BP, you would just use http://servername:port#/ ) >> this would help determine whether the issue is more BP or SharePoint focused.
Other than that, I can't say that we've seen this type of issue with Business Portal, as normally with BP it again, either works or doesn't work.
Thank you