The moon was closer to the earth in the past and it does get a little further away from the earth every year. Just a few centimetres mind, so its progress is very slow. It would not be noticable even over thousands of years to the eye.
It can be measured nowadays though by reflecting laser beams back from mirrors left on the surface by astronauts.
The reason? Well a little energy from the rotating earth is transfered to the moon by tidal effects.
The extra energy makes the moon go a tiny bit faster and creeps it into a higher orbit.
Will the moon ever dissapear. The simple answer is no. Not in the lifetime of the solar system. The outward spiral is just too slow.
If the moon were to dissapear though, the most obvious thing that would happen is that tides would become much smaller. Only the sun would have a significant influence.
There would be no more eclipses of the moon.
Fainter stars would be easier to see in the sky at night for some parts of the month.
Living things that use moonlight as a guide might get confused!
The tidel bulges produced on Earth by the moon are not centered at a point on Earth's surface directly below the moon, but it is shifted ahead of the moon because of the Earth's faster spin of rate. The gravititional force of the excess massin this displaced bulge pulls on the moon, slightly increasing its orbital velocity. Kepler showed that when an object's orbital velocity increases, its orbital radius will expand thus the Moon is slowly receding from us.