Strictly speaking, the tension isn't a force. Force is a vector, tension (or more generally, stress) is a tensor. The Cauchy stress tensor measures the forces applied across an arbitrary flat surface element dividing the rope. In a static situation, the force one side applies to the other is balanced by the force the other side applies on the first, but these forces cannot be added together (and thus cancel out) because they are talking about forces being applied to different bits of the rope.
(It should be noted, when talking about the effect of forces on the centre of mass of a compound body, forces can be added because they're being applied to the same thing. Force is the rate of change of momentum of some defined body or collection of bodies. The momentum of the centre of mass is a single thing. The momentum of the two halves of a rope are separate things.)
The net force on any bit of the rope is zero, because all the forces are being applied to the same object, and thus can be added. The net stress (or tension) on any part of the rope is non-zero, because every bit of the rope applies non-zero forces to each of its neighbouring parts.