A wave on a string is like a harmonic oscillator (think mass on a spring). The oscillation happens because of the interplay between the restoring force, which tries to bring the mass to equilibrium, and the inertial quality of the mass, which tends to overshoot equilibrium. If you take away either, you don't get an oscillation.
If you work out the problem, you will find that the velocity of the pulse is proportional to the square root of the tension. If not under tension, there is no restoring force to counteract the inertial one. All you get is a moving string, not a pulse.