You can set search alerts in many library databases so that you are automatically notified when a new source that meets your search criteria is added to the database. Below is a list of tutorials for various database vendors and Google Scholar.
Some journals allow you to sign up for alerts when a new issue is available. This information can usually be found on that journal's website and will vary depending on the journal's publisher.
More information on staying current can be found here:
Just as you should check to see what a source has cited for additional sources related to your topic, you can also identify where a source has been cited since it was published. These features are not 100% accurate, but they can still be useful.
Google Scholar results can be a little messy, but it does a decent job of identifying citations in the humanities.
More information about cited reference searching can be found here: