We have Umbraco set up on a development server and a production server. When I upgrade it do I need to upgrade it on both servers?
If you are running your sites on a shared database (which I doubt since that would mean any changes on your development setup would also affect your production server - and the other way around) - I would suggest you upgrade by upgrading the live site and then afterwards you do a backup of this to use as development. This is not a good approach since you could risk breaking your production site during the upgrade if anything unexpected happens.
If they are completely separated sites not sharing the same database, you should upgrade them individually. Development first to see if there's any issues - find out how to resolve these and then you're ready to follow the same procedure for production, hopefully without any unexpected issues showing up. This is the recommended approach.