Can the gravitational pull of Earth drag the Moon into Earth's atmosphere? Likewise, can the gravitational pull of the Sun drag Earth into the Sun's atmosphere? If that can happen, what will happen to Earth when the Moon enters the atmosphere of Earth?
You have asked 3 questions. In order:
Can the gravitational pull of Earth drag the Moon into Earth's atmosphere?
As currently configured, no. Way back, the Moon was much, much closer to the earth than it is now, and it has been receding ever since. The effect is well-understood, and it's not going to change.
Likewise, can the gravitational pull of the Sun drag Earth into the Sun's atmosphere?
At first glance, the answer would seem to be no. Not only does the same phenomenon (tidal bulge transferring angular momentum to the orbiting body), there is the non-negligible effect of solar wind constantly pushing outwards on the Earth. In the absence of any major change, this will continue as long as the sun shines.
However, there will be a major change, when the sun goes red giant and expands to engulf the earth's orbit. At this point frictional losses will cause the earth's orbit to decay and plunge to its doom. Per http://arxiv.org/pdf/0801.4031v1.pdf, this will occur (given current models) about 7.59 billion years from now. Of course, by the time the final plunge occurs the Earth will be pretty much hosed anyways as a result of being immersed in the Sun's atmosphere.
If that can happen, what will happen to Earth when the Moon enters the atmosphere of Earth?
If something were to cause the Moon to approach the Earth very closely, the atmosphere will be a very minor player. The reason: the Earth's atmosphere is far too thin and is not a major player. You can do some Googling, and you'll find that 99% of the atmosphere occurs at an altitude less than 30 km. The diameter of the Moon is 3450 km. So the Moon could never "enter the atmosphere" in the sense that I suspect you're thinking of.
Not as things currently are. The gravitational pull of the earth is already pulling the Moon as much as it can, and the gravitational pull of the Sun likewise is pulling the Earth: that is why they orbit. There's no way for gravity to pull any harder than it already is, as long as the Moon, Sun and Earth all keep their current masses and relative positions. In order to change the orbit, it would be necessary to apply large amounts of another force besides gravity on the Moon or the earth, such as from a large object colliding with one of them. However, the amount of force needed would be so great that it would actually break the Moon apart instead of just pushing it as a whole.
Despite this, if the Moon did somehow end up on a collision course with the Earth, before it even hit, there would be catastrophic effects from tidal forces. There would be huge tides in the water, but tides also affect the earth and that would become apparent as the Moon approached the Earth.
The Moon is currently drifting away from Earth. The Earth is moving away from the Sun at 15cm/year.
If the Moon were to somehow come closer to the earth through outside forces, then yes. Gravitational pull, if strong enough, will always pull two objects closer within a certain distance. (But currently, that is impossible.)
If the Earth and the Moon were to collide, the results would depend on how fast the Moon was moving in relation to Earth, but it would be the difference between complete destruction, and complete destruction with a large fry.
Earth's gravity is 9.8 meters/second (acceleration). The Moon's is 1.6 m/s. Force is equal to mass*acceleration. If you take the time to calculate that, you get a whole lot of pain for both involved.
Technically gravity pulls from both objects, so you'd be looking at the mass of both the earth and the moon being pulled by the sum of both gravities... This is more force than in all the nuclear weapons on the planet, and then some.
The answer is no*, on several counts. First, as noted previously, the net effect of tidal heating is to cause the moon's orbital distance to slowly increase. Second, if you could somehow gradually decrease the orbiltal distance, it would disintegrate when it came within the Roche limit: http://en.wikipedia.org/wiki/Roche_limit
Note that one leading theory of the moon's formation involves the Earth being struck by a Mars-sized object.
*Barring indistinguishable-from-magic technology that can move the moon around.