Is there any point on earth where the moon stays below the horizon for an extended period of time? When I was teaching a class why the earth has seasons today, I mentioned how the poles of the earth experience months of daylight and darkness. Then one of my students asked whether the moon disappears as well.

Using an orrery and a string, we looked at the orbit of the moon around the earth and tried to find a place where the moon was below the horizon for an entire orbit. In our attempts, we could find no such place, though some students speculated that if you were in the right place at the right time, the moon could make a full orbit while partially obscured by the horizon.

Is there a place on the earth where the moon cannot be seen on for long periods of time, like the sun in northern latitudes? Or does the fact that it orbits around the earth make this impossible?
Depends on the interpretation of your question...
The best places not to observe the moon are the north and south pole. On the north pole you will only be able to see objects above the celestial equator. As the moon orbits the Earth in one month its orbit is inclined from the celestial equator. This inclination is almost the same as the inclination of the ecliptic (path of the Sun) with the celestial equator. The ecliptic crosses the equator at two opposite points on the celestial sphere. This means that for about half its orbit, the Sun, and as the moon's orbit is near the ecliptic, also the Moon, will be above the ecliptic and therefore visible from the north pole.

That being said, the Moon does not follow the ecliptic precisely as the moon's orbit is inclined from the ecliptic by about 5°. The inclination of the ecliptic is 23°, so during very special circumstances the maximum altitude of the Moon above the horizon on the North pole will be 18° during one month. The duration that the Moon will be above the horizon on the north pole will be about 10 days (a guess) about 13.6 days (edit:see comments below).

So if we interpret your question as: Is there a place where the Moon will be below the horizon for a long period of time (> 1 month), then the answer is NO.

But if this happens near June, then the Sun will also be above the horizon at the north pole (for six months), and as the Moon will be close to the Sun (as it follows more or less the ecliptic), it will be very hard to see the Moon during that time. So if you specifically ask whether the Moon will not be visible for an extended period of time, then the answer is YES.

And there are of course also places with perpetual cloud cover ;-)