Let's say we have a massless body connected to 2 strings. Now lets pull the body with a force $F$. As the body is massless tension in both strings will be equal. Let that be $T$. Now we know that $$F-2T=Ma$$
But as $M=0$, we get $F=2T$. Now although the forces are equal the body still has an non zero acceleration $a$. How can this be possible, even theoretically?
The fact that the equation doesn't tell you the acceleration is 0, doesn't mean it implies the opposite. If A doesn't imply B, that doesn't mean that not B is true. It might be that B is still true but you just can't infer it from A. In this case A is "the equation is satisfied" and B is "the acceleration is 0". The correct way of inferring that the acceleration is 0 is by knowing that massless objects always travel at the speed of light so one cannot accelerate or decelerate them. So B is indeed true, but it could not be inferred from Newton's laws, since they don't apply.
There is no way you can tell if the acceleration is 0 or not from the second law $$F-2T=Ma=0$$
The acceleration is indeed 0 but to prove it you have to use another equation that involves the acceleration or maybe the velocity $v$ separately from the mass. This second equation will then add a constraint on the values of the acceleration.
You have not defined how you are applying the force, nor what is restraining the two strings so as to have tension occur. Force is a vector, not a magnitude. Ditto for Tension. Put in the directions of all these forces to see the answer.
