Burst mode shooting in iOS 7 captures a series of quick-fire photos, allowing you to grab that perfect shot that you likely would have missed if you were taking photos one at a time. When you are done shooting, you are left with a single entry in your Photos app that contains all the photos in that burst. In just a few steps, you can convert that series of shots into a delightful animated GIF.
The first thing you need to do is to select all the photos in the burst shot and extract them to your camera roll as individual shots as follows:
Open the Photos app and find your burst mode shot
Tap a thumbnail then tap "Choose Favorites"
Swipe through each shot and tap the circle at the bottom right of a photo to mark each one that you want to keep
Tap Done and choose to keep everything if prompted
After this step, all your burst mode photos should appear in the camera roll as individual shots. These photos then can be imported into your favorite animated GIF app and tweaked as needed.
For animated GIFs, I use Giffer because it allows you to quickly and easily import photos into the app and has options to create reverse and looping yo-yo GIFs as shown below. Giffer is available as a free app with basic features and a paid Pro version that supports bigger and longer GIF sequences.