Scenario: The Moon disappears slowly -- within 3 days. It is a result of natural causes and we don't have any control over this disaster.
What will be the Major Effects On Earth and life on Earth?
Tides will still exist from the sun's pull, though they would be smaller. Solar tides are 44% of the current total and lunar tides 56%. Tides would occur at roughly noon and midnight every day. Probably some impact on life, weaker tides would stir up less nutrients, etc.
Due to how tides are a drag on the earth's rotation, the earth's spin is gradually slowing down -- days are getting longer. Without the moon, this slowdown would be even more gradual. This effect is not noticeable unless you consider thousands of centuries though.
The earth and moon co-orbit about their combined center of mass which is still inside the earth. No moon means that the earth would lose this wobble. Hard to see any impact on life from this.
Nights would be a great deal darker -- at least compared to those nights other than new moon. Even at new moon, the moon is still a significant source of nighttime illumination due to earthshine. Human eyesight perceives it as dark, but some animals can see quite well in this light. With only the stars and planets nocturnal animals would have a more difficult time, however they would not be as blind as you might suspect.
Brightness is measured in magnitude, lower numbers are brighter. The scale is logarithmic because vision response is approximately logarithmic. These are average figures
Sun: -26.74 Full Moon: -12.74 New Moon: -2.50 The integrated night sky: -6.5
So, the new moon is considerably darker than the rest of the sky combined. In terms of total light, the new moon is about 2.5% of the total. It is still an important contributor, though less than you might think because it is not diffused over the whole sky and still appears relatively bright.
Because of cloud cover, many nights are much darker than they would otherwise be. Since nocturnal animals are used to these conditions as well as the nights when the moon is below the horizon, I can confidently say that though without the moon it would be more difficult to see at night, the animals would still get by, though it might be a little more of a struggle -- every bit counts in a hunter/prey situation and they would never get the bright nights.
Some animals would be adversely affected though as they restrict their activities on the especially dark nights and are more active on the normal nights. Other animals are the reverse, they avoid the full moon and are more active on the darker nights.
Some animals also appear to respond the moons cycle in unexpected ways, e.g., spawning only on a full moon. Clearly, this could be potentially disastrous for them.
The earth's axial tilt is about 23.4 degrees and varies over the course of thousands of year between 22 and 25 degrees. The missing moon would destabilize the earth axial tilt. Over millions of years the tilt would vary from 0 to 45 degrees, some sources suggest that the range would be 0 to 85 degrees. In either case, very massive climate changes would occur. Due to fact that the moon in receding, this process is already very slowly underway -- at least hundreds of million years to be really noticeable. The effect of solar changes is expected to be a much bigger problem first as is becomes hotter over time.
Earth would lose a convenient source of building raw materials for future exploration.
Some suggest the the moon and the human menstrual cycles are related. Given that other mammals have different periods lengths, this is likely no more than coincidence, but we really don't know for sure.
Romance will take a hit. And the ever popular eclipse will be no more.
Finally -- werewolves would not be nearly as much of a problem. At least, no more full moons to set them off.
It occurred to me later, that there will also be a world-wide panic as the moon disappears. Certainly, anything that makes the moon disappear will be very energetic. Just to knock the moon out of the neighborhood, i.e., reach solar escape velocity you would need around 3.8E28 Joules or 9.1E18 tons of TNT worth of energy. That is a 1 megaton bomb every second for 288 thousand years worth of energy, ignoring the fact that Orion pulse rockets are not perfectly efficient. I did not use the Tsiolkovsky rocket equation though. Since there are details I don't know (such as nature of the event) and given the ridiculousness of the answer I figure I am close enough (solely considering kinetic energy change required). I certainly remember laughing at the intro of Space 1999.
Earth's wobble around the earth/lunar barycenter is actually a minor consideration. The barycenter is 4641 km from the center of the earth. The earth orbits around the barycenter once every 27.322 days (the lunar sidereal month). This means the earth orbits the barycenter at about 44.47 km/hr -- 4641 km radius * 2 * pi / 27.322 days / 24 hours/day. The earth orbital velocity around the sun is frequently given as 30 km / sec or 108,000 km / hr -- this number is slightly high, but close enough. Removing the moon instantly would thus change the orbital velocity by about 0.041 percent -- as a vector quantity it could be faster or slower as well as in a different direction.
The earth's orbit already changes by a similar amount in natural orbital variation. Depending upon where the moon was in its orbit when it disappeared the earth's orbit could either become more circular or more elliptical. A more elliptical orbit would result in more climate variation, but the earth orbit would not be quickly destabilized by this catastrophe. Since the earth's orbit is more circular than average at the moment, the resulting orbit would still be similar to natural orbital variation. Over millions of years, the effect is not predictable (n-body stability is unproved) and you get into the realm of butterfly effects. Over hundreds of years Earth would be essentially unaffected in terms of a stable orbit because of the sun that is so massive compared to Earth.
I failed to include this effect and Dan Smolinki did, but I am certain he overstated the impact re: orbital stability. Definitely a good catch though as greater climate variation would be a bad thing.
ADDED -- Now with catastrophes
3 days to disappear is certainly not slowly disappearing in my mind, but I think there are other effects that I did not consider before.
Lunar tides means water is pulled up by about 0.4 meters / 1.5 feet in mid-ocean (considering just the lunar component). If the moon disappeared instantly, that would likely result in fairly large tsunamis at the shore at this kind of water displacement is similar to that caused by major earthquakes.
If the moon accelerates away from the Earth relatively slowly, this would spread out the water displacement effect over time. Considering that lunar tidal effects result in 2 sets of high/lower tides every day, there would be little change noticed from the moon leaving slowly.
If the moon departed to do a sudden velocity change, the effects would be intermediate, but unlikely to be dangerous. It is really the near instant change in the water displacement that would cause large tsunamis.
The "natural causes" for moon leaving in 3 days are fairly limited in terms of likely causes, the most obvious being an impact or near miss event. Impact would be catastrophic for Earth as debris rains down on the Earth over time (many years in fact), but the heaviest debris would start within hours of the impact.
Something massive enough to de-orbit the moon in a near passage could also affect Earth orbit noticeably as the passive object would have to be very massive -- did not do the math, but I would expect the passing object would have to be considerably more Earth mass to steal or simply de-orbit the moon in a single quick pass -- a wandering black hole being the likely candidate.
Since Earth's mass is 81 times as much, the delta v would likely be 81 times less +/- a considerable factor as the distance at which it passed Earth could be considerably larger or smaller than the distance from which it passed the moon. For example, earth moon distance is about 385,000 km if object passed 38,500 km from moon, the gravitational effect would be 100 times stronger. Thus the net effect on Earth orbit could be relatively minor or major depending upon the particulars. Of course, we would detect such an object well in advance due to the gravitational anomalies and we could begin the world-wide panic before the actual loss of the moon. End of the Earth riots could cause major damage even if the black hole passage does not by itself.
But now we have to consider that our wandering black hole would result in some seriously strong tides that should guarantee some pretty catastrophic tsunamis and floods. The crustal strain would probably trigger some earthquakes and volcanoes too at a minimum.
In any case, those disappointed by the lack of catastrophic effect can take comfort in having at least some catastrophes deriving from the direct effects on Earth that would occur at the same time.
The moon is in fact important because it stabilizes the Earth axis. Without the moon, the Earth Axis would slowly (we are talking about geological timeframes, thousand of years) change their orientation, making life on Earth much more difficult:
Laskar,Joutel,Robutel: Stabilization of the Earth's obliquity by the Moon
Muslims would be quite aghast because their calendar, the Islamic Calendar is based on the moon. The beginning of a month is defined by the first sighting of the crescent moon after new moon (رؤية الهلال: hilāl ) and the holidays are based on the Calendar.
Losing the moon would perturb Earth's orbit. It would cause it to become significantly more elliptical, and almost certainly would put us in a non-stable orbit.
Over time (possibly as short as 100-1,000 years?), this would likely render Earth uninhabitable. We might end up further or closer to the sun, in a very elliptical orbit that alternately freezes and then burns the Earth, captured as a new moon of one of the gas giant's, or even ejected from the solar system. It's possible that it would stabilize back into an orbit that's conducive to life, but unlikely - orbits are tricky. If you've ever played with a solar system simulator, you'll know orbits are very easy to screw up if you introduce or change current masses.
The importance of moon for the stability of the earth axis should be considered when investigating the prospects of intelligent life in other planetary Systems: While there may be many earth-like plantes in the habituable climate zones, only quite few are expected to have a companion similar to the moon, i.e. quite large. The moon may have contributed to a more stable climate suitable for life and, even more, for agriculture. I would also like to raise the question whether there may be some correlation betweeen the moon and tectonic activities, in the long run.