If one strikes a note (say C3--an octave below middle C) on an acoustic piano without the damper pedal and then with it, one will likely notice that the second note sounds slightly different--probably brighter. This is because the struck string will vibrate with energy not only at the frequency of C3, but also some other frequencies including C4, G4, C5, E5, G5, and C6. If the strings corresponding to those other frequencies are not damped, some of the energy at those frequencies will transfer to those strings, causing them to resonate along with the string that was struck.
If all one ever wanted to do on a piano was play notes that would sustain as long as the strings allowed, having all of the strings undamped would probably improve the sound. A lot of music, however, requires that notes not sound for anything near the durations achieved by the lower strings on a piano. Consequently, the lower strings need dampers to avoid allowing music to be played at a reasonable speed without turning into mud. The upper strings on a piano have a natural decay rate which is much faster than that of the lower strings; additionally, their higher natural frequencies may allow them to add more brightness than would the lower strings. These two facts together have led piano makers to omit dampers from the upper strings (the fact that such omission saves cost probably contributes to that decision, but even expensive pianos generally omit the dampers).