Solution
After some researching and asking around (thanks to @PHPology), I figured out that this was an issue because my blog was marked as "private" (password protected) in tumblr settings.
Make sure your blog is not password protected, else the buttons will not show/work as expected.
Also ensure your HTML is valid and correct and there are no javascript errors (tumblr loads these buttons in dynamically via javascript after page load - Javascript errors will stop the execution of this script and others).
Customizing the like button hover
A great post here on how to apply hover effects to the like and reblog buttons within the iframe.