I have the same problem with my brand new iPhone 6 128gb, just opened the headphones that came included in the box plugged them in and they would not go all the way in. You can clearly see and feel the jack come in and out. Not sure if this is a ploy to have you buy bluetooth headphones, but I feel the ones that came with the phone should work without a problem. I have been a loyal Apple follower over iPhones, iPods, Macbook Pro, Apple TV, and iPad with no issues. Did some research as well as read the directions to make sure I was not doing anything wrong, maybe there was a trick I thought I was missing. No such luck, but I did stumble upon this board with multiple people experiencing the same issues. I could not even use my phone to run with as the jack kept coming dislodge just a bit causing the song to stop while when it was working I heard a constant popping in my ears as the jack jostled about.
I hope there is a simple fix to this or maybe an issue in a small batch of phones, I love the new phone in every other way but the phone should work flawlessly with the included accessories!