djamal , I guess the meaning is totally different from حساس؛ أو عاطفي , as "He has a huge chip on his shoulders" , means he is someone who is ready to get in fight if anyone touched him (he is about to explode).
In Egypt (and I guess also in Syria and Lebanon) we say " بيخانق في دبان وشه", which means he " he is very angry that he fights any fly that comes near by him "..
We may also say describe him as "شايط" or "مفروس" .. and as expression you may say "محدش يجي جنيه اصله شايط / مفروس " .. which means "don't touch him he is about to fight .."
we may also say " مش طايق نفسه " or " مش طايق حد يكلمه " which literally means "he can't stand himself " or "he can't stand anyone" .
That is not completely right as well. To have a chip on one's shoulders is to bear a grudge or grievance over something and to take offense easily. When the subject comes up one may get argumentative and/or belligerent, but that, to me is only a secondary connotation of the word. The main idea behind it is that one is bearing a grievance. I cannot think of any equivalent phrase in Arabic. The first thing that came to my mind, though, was شايل من الدنيا shaayil min id-dunya. But this just means to have it out for the whole world.