Tumblr doesn't have a built-in feature that enables you to check whether your followers are logged on or not. However, you can get a general idea of when they're active by keeping tabs on your live activity notifications and monitoring follower engagement metrics.
View Live Activity
By watching live activities you can get a general idea of who's currently online. Log into your Tumblr account and open your dashboard. Click on "Activity" and examine your current notes. This page shows follower events such as Likes, New Follows and Reblogs. You can receive real-time email notifications about activity by clicking on the "Settings" icon, "Email" and checking off "New Followers," "New Replies," and "New Messages." Additionally, the Tumblr mobile apps for Android and iOS will send you push notifications automatically whenever your followers like or reblog your entries.
Free third-party analytics tools such as Numblr, Google Analytics and Simply Measured (links in Resources) enable you to view time and demographics data about your Tumblr visitors. This can help you gain a general idea about the peak traffic times on your Tumblr -- when your followers are most likely to be browsing. Numblr and Simply Measured require that you sign up for an account, and then they display metrics on their Web apps. Google Analytics requires that you paste a custom HTML Web code into your Tumblr theme to collect visitor metrics.