Nothing serious happens, but you can't proceed to the Elite 4 without all 8, so you will have to go back and get the badge you missed.
What may happen is that you will play the scene after the 7th Gym (i.e the attack by team plasma in BW2) before the scene after the 5th gym, meaning the order of events gets a little screwed. But it's not serious and won't do any damage to your game, but it can be annoying.
Source: Experience
