Apple - How can I get an iPhone to play songs in order

Is it possible that you have shuffle enabled? The shuffle button in the playlist list is not a toggle, so that wouldn't tell you. Here's how you can find out:

If the shuffle icon is blue, it means your iPhone is playing on shuffle.

Something else you could check is if Shake to Shuffle is enabled:

If it is, you could accidentally activate shuffle without knowing it.

Hope this helps.

Since iOS 8.4, you can play songs in sequence by clicking on the 1st song; when the song title is showing at the bottom, drag it up and you will see the controls. Unlock the shuffle button by clicking on it.