It's tempting to make a joke about what a "harmonic dampener" might do (spit on you while singing?)
A torsional vibration damper is used to limit the amplitude of the torsional (twist) oscillations of the crankshaft.
The oscillations are excited (caused) by both the firing pulses and inertia loads transmitted to the crankshaft by the reciprocating assembly (and to a much lesser extent by the timing drive and/or fuel system for high pressure injection). These excitations are made up of many harmonic components. The crankshaft+flywheel+rods system has a number of different mode shapes that it would like to take while vibrating, each with its own associated frequency. Crankshaft vibration is greater when one of the harmonic components of the excitation is occuring at the same frequency as one of the crank's vibratory modes (when resonance occurs).
Torsional vibration dampers come in four main varieties:
1) mass-elastic-damper <- typical rubber damper design
2) viscous damper <- seen on large engines, also available in aftermarket for autos
3) visco-elastic damper <- seen on large engines, probably available in aftermarket for autos
4) mass only (absorber not damper) <- seen on aircraft engines, available on aftermarket ("Rattler" brand, for one, I believe)
Type 1 must be tuned (correct stiffness, damping, and mass) to work well with the crank system. It will often be selected to split a big resonance hump in the running range of the engine into two smaller humps, perhaps with one of them above the running range of the engine. It is not unlikely that changing from a cast iron to a steel crank without changing anything else will not have a large enough effect on the crankshaft vibratory mode frequencies to make a huge difference in damper tuning. You could no doubt do better with a custom-tuned damper, but the original might be good enough.
Type 2 is "sized" rather than tuned. It simply absorbs vibratory energy from the crank, and doesn't have the "hump splitting" effect of type one.
Type 3 must be "sized" and tuned, but is more similar in behavior to Type 2 than to Type 1.
Type 4 is a special case - it doesn't really absorb energy the way the other three do - it produces a counter-excitation at a specific engine order. It's very useful when you have one particularly annoying excitation order to deal with, such as when you're running a fixed-speed engine.
There are lots of reasons why some motors can run without the dampers; I don't know which might be the case with sprint cars. If an engine runs at speeds that put it below or above its major resonance humps, then a damper may be unneeded. If the crank system mass has been significantly reduced, it's possible that some of the vib. frequencies would shift out of the running range. If the crankshaft is strong enough that it won't break anyway, then damping the vibrations would be unneccessary (from the point of view of the crankshaft stress, anyway). If the crank life requirement is short enough, then you might allow damaging vibrations to occur with the expectation that you'll replace the crank regularly.