Since the branches are dying, I'd cut them off. You will have to cut them off sooner or later anyway. Better to do it now. Also, if there is a tip weavil or beatle infesting the tree, cutting off the affected branches can stop further damage. I don't know how far up the dead branches are, so it might disfigure your tree. It could be that the tree is simply at the end of its life, too. But I'm just speculating. In any case, prune out the dead and dying branches and see what happens in next couple of months.

Once you cut the branches off, take them to your local greenhouse and see if they can tell you what caused them to start dying. It helps to put your mind at ease if it's just age or drought or whatever.