Removing posts from canals could be pain in the neck sometimes. There are many ways to remove a stubborn post from a canal such as cutting all the sound tooth structure around the post with a large round bur (I hope nobody is using this method) or using ultrasonic device and frying a few expensive tips in the process and ultimately the root and surrounding structure.
One method that has allowed me to remove tough posts more conservatively and relatively quickly is using the Ruddle Post Removal System (PRS) Kit.

The kit comes with a series of post removal trephine burs (#1 to #5) and corresponding post removal tubular taps. The process of port removal starts by removing the majority of the build-up material from around the post. Then a trephine bur can be used to mill the post to a specific size (usually the biggest size that starts to mill the post). The corresponding tubular tap is then used in counter-clockwise direction to engage the post. The tap is screwed in to the point that it locks and starts to disengage the post from the canal. If the post cannot be removed easily, the extracting plier and cushions will do the job beautifully. The short video clip below demonstrates how easily a post can be removed from a canal:
Dr. Mahmoud Ektefaie @ vanendo
What do you do if you have to remove carbon fiber posts?
Carbon fiber posts are easily shattered by ultrasonic tips such as ProUltra Endo Tips 3 and 4. You can ultimately use straight diamond bur (surgical length) to drill through them. In either case, you need good magnification, light and lots of irrigation so that you are certain that you are chasing the post and not going sideways.
Can you use the Ruddle Removal tool as a leverage device on a threaded post or will this fracture the root due to the engaged threads? (Assuming it wont thread counter-clockwise as shown on the video)
In my experience, the threaded posts unscrew easily with this device. The only couple of times that I had to use the extracting plier was when I had to remove long, non-threaded, parallel and very well-adapted posts (i.e. no sealer or resin evident on either side of the posts). Since the extracting plier applies force in the direction parallel to the long access of the tooth, I did not notice any issue with tooth or root fractures during the post removal process.