If you remove the limb, in a few years you'll be asking, "What do I do about that big hollow spot on the trunk, remove the whole tree?"
So no, you don't want to remove the branch.
Xtremetrees called it a parasite branch, but branches are not any more parasitic than roots are parasitic.
Thinning was mentioned. To me that would just be a feel good action. If it would make you feel good then do it, the tree will most likely survive it, I don't think it would help one bit. Removing a small percentage of the crown, even if it's just on one side branch, will slightly change the wind resistance, but it could be argued whether it increases or decreases that wind loading. Trees tend to take wind loads as one single mass, unless they are wide and multi stemmed. Cutting holes can have the effect of increasing wind resistance on the main spar.
I won't cut healthy living branches off a tree without a very good reason. Mature trees reach a balance. Changing the amount of food a tree can make by cutting off the branches, can throw a mature tree into a downward spiral. At the very least, it's a stress to the tree.
Topping was mentioned, which I can't believe, given the amount of information about how bad a practice this is. Shame on you Ekka. A google search resulted in 2,030,000 sites that say topping is bad. Do the search if you need more info on that.
Cabling was thrown up a few times. Steel cable would increase the likelihood of failure, due to the fact that there isn't a weak spot to protect, and cables can cause failure by karate effect.
Dynamic cables would be a great feel good action. They hold the limb to the main trunk and don't cause other problems.
If it were my tree, at my house, I'd do nothing but sit back and enjoy it.