This is a guide on how to download Flappy Bird on your iPhone, even without jailbreaking it, now that the game is gone from the App Store.
For this to work, you will need to know someone else who has an iPhone and managed to get Flappy Bird before it was pulled from the App Store. They also need to be willing to give you their iPhone account login. Assuming you have all of that covered, here is what you do:
On your iPhone, go to the App Store, sign out of your account, and then sign with your friend’s account. Go to Purchased Apps and you will see Flappy Bird in there. Make sure to look under the All Tab. Even if your friend deleted their copy of the game, it will be under that tab if you check. Once you have found it, install it, and then sign your friend’s account out.
When you sign back in, the game will still be in your iPhone. You can then sync your iPhone to iTunes, so you can keep Flappy Bird in that account.
Outside of this, your options are to jailbreak your iPhone, or, perhaps sensibly, get one of the many Flappy Bird copycats. Some of them are arguably better than the original, anyway.