We have been using Tableau Server for 2 years. Tableau is very generous with their Server licensing and allows you to have up to 3 identical servers using your license model. So we have tableau-production, tableau-dev, and tableau-beta. At this exact moment production is at 6.1.6, dev is at 6.1.7, and beta is at 7.0.1.
We take our nightly user and data backups off production and restore them onto dev and beta so they're all in sync, which also gives us a superior high availability option because if production ever goes down due to hardware, software, or licensing failure, users can be redirected to dev.
We apply the minor point release to our dev machine a few days before updating production and we rarely see a problem. It gives IT a chance to download the update and practice so they know how long it will take. There have been a few cases where we've had to hold off or even rollback a minor release for definite bugs. But you can always do that to your production server and your workbooks will be fine.
Now for the major updates like 7 we were in the beta and had always planned to wait until at least the first point release after launch and we're still trying to tune the new server so we're not they're yet. But when we do upgrade we will first upgrade Server to 7.0 and then a while later start upgrading Desktop for users. That allows you to roll back in an emergency, which you can't do if your users start publishing in the new version. Back in the days of Tableau 5, we upgraded Server to 5.2 but never upgraded Desktop beyond 5.1 because we didn't need maps.
So in summary you should always apply the point releases but the major releases you need to plan.