Technically speaking, only the Moon would crash into the Sun. That being said, it's certainty one of many possibilities in the far distant future.
About 7.6 billion years from now, the Sun will have ballooned to a large enough size to engulf much of the inner solar system. One of the consequences of growing into a small red giant will be mass lost due to a rapid sheding of material through a stronger solar wind. This mass loss will allow all of the planets to be pulled on by the Sun less, and they can achieve a higher orbit. In Earth's case, it could be as much as 1.2 AU from the center of the Sun. Also, there will eventually be a large increase in the solar wind density where Earth orbits the Sun. This solar wind and chromosphere will create drag on Earth while in motion around the Sun, and will rob the planet of its angular momentum (orbital velocity). The Earth will slowly begin to transition into a smaller orbit, and some sources think that this may lead to the Earth falling into the Sun.
In this scenario, Earth-Moon system is gravitationally stable, and the Earth has sufficient gravity to hold onto the Moon during this time - Earth will pull the Moon down with it.
What is not known to me is if the drag from the solar wind and chromospher will cause the Moon to lose enough angular momentum to spiral inwards, ultimately merging with the Earth before the Earth is engulfed by the Sun. This would be a spectacular sight, as the moon would first be ripped apart by the tidal interaction with Earth (inside the Roche limit, or about 11,500 miles from the center of the Earth). Our Moon would resemble a string of moonlets, each eventually falling to Earth. Either way, the result is the same - the Moon will only exist within the Sun.
Sources:
http://www.space.com/3373-earth-moon-destined-disintegrate.html
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