Well, might be a bit hard to help me since I don't know anything about linux but please give it a try 
I'm paying for a vserver, preconfigured with debian 4.0, and since this machine has not a lot to do I thought it would be nice to host our guildbot on it. That was the easy part of it...

Actually I think I did quite a good work until now, apache, php5 and mysql are working, bebot 0.4.3 is installed and working, even under different user than root (aorunner) and no superuser-rights needed.

My problem is:
I do need to have an open connection to the vserver to keep the chatbot running, as soon as I close the shell I'm using to start the bot it is shutting down. No message, just going off.
Is there a way to "install" the bot as a background task or how will I be able to keep the bot running without an open connection?

There is either a script flying around here that starts bebot as a background service on linux.

Or install the tool "screen" as Dracutza said, it allows you to detach standard-input and -output from the normal shell. Basically something like the following:
Then do CTRL+A D to detach.
To re-attach you only need to do screen -r.

Thank you all, especially for the detailed howto

I found the script before posting here but it didn't work for me, in the thread is mentioned to start the bot with a perimeter ('php StartBot.php &') but this didn't work either.
The screen tool is working fine

@Dracutza
I did several forum searches before but they didn't show up any result according to my problem. But I'm still wondering how I should have come to the idea to search for "screen" or "screens", as I said I'm new to linux and I never heard of this tool before.

A search on "Linux" got me the screens data