In order to use the Follow, Reblog, and Like features of the API you will need an authenticated user, which requires the end user to grant "access" to your API application.
In other words, yes, such a "pop up" will appear asking the user to give your site permission to "read and write" to their Tumblr account.
As workarounds, that are not quite fully integrated but will usually work OK:
To add a Follow link, include a link to something like:
http://www.tumblr.com/follow/{Blog Name}
To include a Reblog link, include a link something like:
http://www.tumblr.com/reblog/{Post ID}/{Reblog Key}
The "reblog key" can be obtained in the post data obtained from the API.
As for the Like button -- you're just never going to get that to work without an authenticated user (ie, going through the "pop up" and obtaining private/public OAuth keys to access the user's tumblr account)
Edit:
However, remember that inside your Tumblr theme code you can use {LikeButton}
. This will render an <iframe>
element provided by Tumblr that implements the "Like" feature. It would be possible to load pages from your Tumblr with AJAX and parse out these <iframe>
elements and insert them where needed dynamically.