Does archiving email remove it from an exchange server?

Outlook will prompt you to do this whether or not you are low on space. Archiving moves emails to another folder in a static .pst file stored locally (network stored pst files give huge performance problems and can cause more crashes, so not worth it - stick with it being on your local drive and make a backup of it)

If you archive or even delete emails, they may still be left on the Exchange server, as this can be configured to keep emails until the next successful backup, for example. Likewise, on Exchange 2003 and earlier (and I'm fairly sure the same applies in 2007, but not 2010), if an email was sent to two or more users on the same mailbox server, then the email is stored only once, along with a list of who still has that email. Only when everyone has deleted or archived off the email will the email actually be removed from Exchange and save any storage space. Even though it would not longer count against your quota you may not get any more space available to the Exchange server.

Also, if you delete or archive emails and you are the "last on the list" and it is deleted, then a backup is taken and the email is really gone, this only frees up space in the message store, this does not reduce the amount of disk taken up. It will mean that the free space can be used first though, which will slow the increase in store size for a while (depending how much you get rid of and what volume of mail the server handles).

Archiving in Outlook basically means that all e-mails matching the criteria for AutoArchiving are either deleted or moved to a separate .pst file. This file (usually called "archive.pst") is stored on your local hard disk. The archived e-mails will be removed from your Exchange account because that's the whole point of process.

Microsoft has a article which explains AutoArchiving more detailed.