There are sort of two parts to that question. One part is that the height above ground affects the amount of reactance between the antenna and ground. That means that tuning the antenna can change if raised or lowered an appreciable amount.
Another part is that height affects the radiation pattern. The horizon gets further away the higher the antenna is, so it can 'see' further in a straight line, before any 'skip', sort of. Not too important at HF really, but it is important at VHF/UHF. So, the higher the better to some ridiculous point.
So, how low can an antenna be? The lowest for most vertical or groundplane, or non-horizontal and directional antennas (beams) is -at- ground level, does no good at all below ground level. Those directional antennas have sort of a catch, cuz they could be 'vertical arrays' such as broadcast stations have. Ground level and directional, sort of harder to turn though.
Best all around height for a typical vertical groundplane type thingy is as high as you can get it.
The only kind of interference that height can 'cure' is 'over-load', since it's further away from the affected device(s). Won't 'cure' spurious or harmonic type interference since antennas don't 'produce' any signal, they just radiate what's fed to them. They can attenuate or increase the signal, just can't make any of their own.
- 'Doc