I am a big fan of CI (continuous integration) and automated tests, but please don't do this:
> Even better: continuous deployment! When the tests pass, deploy.
Continuous deployment (CD) means deploying to your production environment. This is dangerous and should always be a conscious decision by someone (preferably even multiple persons). Normally, a new release either fixes a bug or adds new functionality.
If your using CD and your software has a bug, that means that your original tests were not sufficient. The fact that you now have new software and new tests does not mean that you have solved the original bug.
If you want to add new features, you will also have to add tests that these features work as intended.
In all these cases expert human judgement is needed, it is not wise to automatically deploy changes by developers.