The first on my list of most sensitive parts on a horse would be the head/face. A lot of horses I know don't liked to be touched at all in the face while others seem to love it. Horses' heads can be very sensitive and horses do not usually like to be touched on the face (unless they have a favorite spot they liked to be rubbed). When a horse refuses for their head/face to be touched, it's best to force it. Keep trying to touch their face until they learn that when they let their head be touched, they take the pressure off. It's important that you are allowed to touch any part of them without them feeling discomfort or anger. Of course you have to be careful what you do if it's on a sore part, be alert on the horse's behavior and you can tell what bothers him if he flinches, jumps, ect.
The ears are also sensitive and when putting on a halter or bridle, be careful not to shove them around. Each horse can be different: one part may be much more sensitive on one horse while the other one may love to be rubbed there. Same with a horses' ears, but since you're in charge you must be allowed to touch their ears (if there is any injury, the horse will trust you and know that you are helping, not trying to hurt, them). Also when taking or putting on a halter, you have to be careful of the eyes.
Same kind of thing with the mouth, lips, teeth/gum and tongue. Their tongue is very soft but we put a cold hard bit in their almost every time we ride them. But bits are not made to hurt them (although some bits could) and if we know how to ride well, we won't be hurting them.
The Fetlock (lower part of the legs) are sensitive because they have very little muscle tissue. They also have a thin layer of skin covering the bone. Sometimes the fetlock is wrapped when the horse is riding in a trailer. The fetlock suffers the most injury when the horse is in some kind of violent position. The weight of the entire horse is focused on one leg when a horse is going a particular gait.
The withers can become sore easily when the horse has an ill-fitted saddle. The flank is very sensitive on most horses and when touched some horses will tighten up and might kick when their feet are been checked. But horses are trained to be touched on any part of their body if it's sore/injured, or fine.