What happens if you only work your arms at the gym?
You make no mention of your goal. So, let's assume that it's one that's shared by many people: a balanced, symmetrical, and aesthetic physique. Working only arms will not accomplish that since you are neglecting other body parts. In a sense, if you were an artist, you're painting a small portion of the picture and leaving the rest unfinished. Additionally, while you may get bigger arms, and that's not a given, your potential for an overuse injury increases. And, having an unbalanced physique may put more demands on the other body parts that you have neglected. All in all, working only one body part is never a good idea.