Do any of the posted answers take account of the planet's inertia?
I'm certainly NOT about to be the first poster to say that the Earth does accelerate towards us. Because plenty of the previous posters have said exactly that, while mentioning that in practice the effect would be too small to notice or to measure.
I am going to complain about the answer that the Moon can accelerate toward the Earth, because it is fairly widely known that the Moon is in orbit, and as frame-dragging effects caused by the Earth's gravitational field accelerate any orbiting object, that effect causes the Moon to move away from the Earth at a continuously accelerating rate, which I have a memory from childhood of being a rate of approx one inch per century (or metric equivalent).
If it were in a retrograde orbit, it would at least be capable of decreasing its distance from the Earth over time, since then the frame-dragging effect would be decelerating it, instead of accelerating it.
My actual answer is more down-to-earth -- :)
Since the Earth has a large mass, and since one fairly well established property of mass is inertia, I'm willing to go half-way, and say that the Earth doesn't accelerate toward us: because it doesn't move at all. If I perform the jumping experiment (with a mass of 180 lbs x 1), or even if I get all the men in China to (180 lbs x 1 billion), the Earth is held in place in spacetime by the inertia associated with its mass.
It's approximately equivalent to throwing a tennis ball at an approaching freight train and expecting to derail it, or to halt or delay the train, even temporarily: mathematically, a calculation might be done that demonstrates there is a calculable effect (as some on here have done); but such calculations tend to ignore the inertial component (quite large, for a body of planetary mass).
If I had some significant fraction of the mass of the Moon, and then jumped, I might reasonably expect that a measurable effect would result. But the Earth is massive enough for its position to remain unaffected below a limit determined by Kepler's laws of planetary motion.
Again, I have a memory about Newtonian conservation of momentum, but which I suspect won't apply within a closed system: me plus the Earth.
But in relation to does the Earth accelerate towards the Moon, well the answer is it does! Well, it does in part, at least. It's called the tide, and at any point on the Earth's equator that has open sea, you'll experience high tide once a day when the Moon is more or less overhead.
This is due to planetary inertia! If the Earth had no inertial component to its mass, when the tidewater moved 4 ft closer to the Moon at local noon, so would the Earth: in that case you would not notice a change in the tidal level, because both ocean and seashore would have moved by an equivalent amount.
The fact that you do notice the tide rising and falling each day is a proof of the existence of inertia (the Earth has it, so the seashore has it), but the sea has very much less of it.