Tumblr offers many wonderful features, but an intuitive messaging system isn't among them, as you've most likely discovered to your chagrin. Even relatively simple information, such as the date of messages in your inbox, can be difficult to find -- or even impossible, depending on the type of message.

Asks You've Received

If the message in question is an ask somebody else has sent you, finding the date is not impossible. Click the top right corner of the message and it will open in a new window, shown as if it were posted on the Tumblr blog that received it. The date the message was sent is in the same location that post dates are on that blog's theme.

Other Messages

As of January 2014, Tumblr offers no way of finding out the date of any other message in your inbox. This includes both responses to asks you have sent to others and fan mail. The best you can do is guess the relative age of a message based on the date of those that precede it and follow it, assuming those messages are asks you've received.