First:
You can delete all the files in /var/log that end with a number or serious of numbers. You should not delete any other files in /var/log, if you can help it.
If you do delete the files in /var/log that are NOT followed by numbers they will continue to grow and take up space anyway, until you stop and restart the syslogd daemon.
Of course if you delete any log files you'll lose the information in them; if you feel you may need them later you may decide to move them to a storage directory on another filesystem, but if you do, it'll become your responsibility to maintain them.
Now let's discuss the problem: Yours is not the first server to have been created with a /var partition of only 250 MB.
In order to help other people avoid this problem, can you please tell us:
1) What OS and version you're running ?
2) Did your server come with a 250 MB partition, or did you create it yourself?
3) If a data center did the install it would help to know who they are so we can educate them.
Now that that's out of the way I'll start by saying that it's a serious problem, and that there's no easy way out of it...
Your DA server not only saves it's logs in /var, but also all incoming and outgoing email; on a busy server that can become many gigabytes very quickly, depending on whether or not your clients save any pop3-downloaded email on the server.
For exmaple, i just checked one of our servers; the /var usage is at 2.3 gigabytes.
You can move subdirectories to other file systems, but it's easy to make mistakes, and some subsystems require you stop and start certain services and/or daemons while doing so. And then you'll have a non-standard system that's no longer taking advantage of the reasons you've got multiple filesystems in the first place.
There are programs available that will enable you to resize partitions on the fly, but I don't recommend them for several reasons. Among them are:
1) They're dangerous to use; if they fail for any reason you'll have an unusable and unrecoverable system.
2) Unless the partition you want to enlarge is right below the partition you want to reduce, it can become very hard to figure out the steps you need to take to get the job done right, and you greatly increase the chances you'll end up with an unusable and unrecoverable system.
You can add another drive and do some moving of data and remounting, so you can put your /var partition on another drive. This is the method we use for our clients when at all possible.
You can save your system, rebuild it from scratch, and restore it. This causes a lot of downtime and if you're not careful you can lose some client data and end up spending a lot of time restoring the rest.
Of course if you're renting the system you can just rent another, partitioned properly (there have been a few threads on the subject) and start moving over domains at your leisure. Then stop renting the original server when there's no more customer data on it.
While this is the most expensive solution, I recommend it highly as the best solution for a rented server.
Jeff