For example, in a hanging massless rope, or in a massless rope hanging over a frictionless pulley, the tension must be the same at all points, because otherwise there would be a net force on at least one tiny piece, and then $F = ma$ would yield an infinite acceleration for this tiny piece.
What happen if the tension is different at some point?How it yields an infinite acceleration?
Well, I don't really like this argument. But the idea is like this:
Consider a chain of identical massless springs, connected to fixed points at both ends. Between each spring, there is a ball of mass m. (Ignore gravity.) We'll later take the limit of $m\rightarrow 0$ to make the whole chain massless.
Now consider the force on a single ball somewhere on this chain. If all the springs are taught with the same strength - i.e. tension is equal everywhere along the chain - then the forces in both directions will balance out and the ball will not move. It'll stay fixed in its equilibrium position.
Now consider the case where the tension is not uniform, i.e. some springs are more stretched-out then other. Think of a ball pulled by a lax spring from the right and a more-taught (more stretched spring with the same spring constant) spring from the left. (This is not a stable situation, but it can happen momentarily.) The ball will experience a net force $F_{net}$ to the left, and will accelerate with an acceleration of $a=F_{net}/m$.
Now take the masses of the balls to zero, $m\rightarrow 0$. The acceleration of the ball becomes infinite.
In other words - any section of the rope where there is a change in tension will experience a net force, and if that section has zero mass this will result in infinite acceleration.
Now in practice there is always some change in the tension along the rope, which makes the masses (fibers, molecules, etc.) become tighter or looser thus re-distributing the force - with finite accelerations, as the fibers etc. have mass. All of this pushes the positions of the masses to their equilibrium positions, in which the forces are balanced and thus tension is uniform.
Which is why I don't like this argument. The real reason the tension is constant is not that otherwise we'll have infinite acceleration, but that this is the equilibrium state.
