Tracking a tag on Tumblr can be done only from that tag's page, not from the search results page. Tracked tags appear in a drop-down menu that unfolds when you click inside the search field. This gives you a convenient way of seeing when new posts have been made with that tag, as well as an easy way to reach that tag's page to read those new posts.
As of publication, Tumblr has a soft limit of 20 tracked tags in place. Although you can track as many tags as you want, the new post alerts in the drop-down menu stop displaying if you're tracking more than 20 tags.
Open the tag page. You can do this either by clicking the tag when it appears on a post on your dashboard, or by typing the tag page's address into your browser's address bar. For example: http://www.tumblr.com/tagged/kittens or http://www.tumblr.com/tagged/cute-cats
Click Track This Tag.