I know this tip seems kind of obvious, but I rarely used burst mode on my Fujifilm XPRO-1 and I only just got the iPhone 6 Plus (burst mode is only available on iPhone 5S models and up). I have only recently started using it more and have found it quite useful. Below are some reasons why you should consider using burst mode (in certain situations):
Fast moving sports or even slower moving objects in street photography.
By taking a burst of shots you will have a greater selection of scenes to use, where your subject/s (or most of the subjects) are posed the way you'd like. For example, I used burst mode to capture multiple shots of this swimmer doing freestyle laps at the Bondi Icebergs Lap Pool:
If I hadn't used burst mode, there would be a greater chance I would have gotten a dud swimming shot like the one below, where it looks like he's drowning.
Constantly getting blurry shots?
If you find that you are constantly getting blurry shots, using burst mode will increase the chances of you actually getting a usable non or less blurry photo. Of course it pays to learn how to shoot with a steady hand (tuck your elbows and hands in close to your body) but sometimes hand wobble effects the best of us.
Digital storage is free.
With traditional film cameras, I think it costs roughly $1 a shot after all things considered. Welcome to the digital age where photography is practically free. It doesn't cost you anything to shoot a million photos and so you should take advantage of that and fire away.