Facebook recently began supporting 360-degree photos. However, the upload process was rather quirky and required dragging and dropping on the desktop.
Fortunately, with the Facebook update on July 7, you can now upload 360-degree photos via the mobile app. Just create a new post and tap on the photo icon, and choose a 360-degree photo. When you upload the photo, it will automatically be recognized as a 360-degree photo.
I've tested it successfully on Facebook for iOS (version 59.0) with the iPhone 6, and Facebook for Android (version 184.108.40.206.70) with a Samsung S6 running on Android 6.0.1.