Can I change the email notification sound on my iPhone?

No. Apple offers only one email notification sound to let you know a new email has arrived. If you are interested in changing this tone you must jailbreak your iPhone. You can switch the email notification sound on or off in Settings.

To turn the notification sound off or on touch Settings -> General -> Sounds -> New Mail -> ON/OFF.

I really wish these "tips" were dated. Because this one is dated. I don't have On/Off for my New Mail sound, I have dozens upon dozens of rings. Oh, and gosh, they all just go buzz.

You can actually on/off the sound for incoming mail. You can even change different types of sound for different email accounts. Go to notification centre > mail. Your different accounts will be listed here. Go to one of them, go to "alert sound" and choose from the sound list.