Why doesn't the Earth's acceleration towards the Moon accumulate to push the Earth off its orbit?
Because the Earth doesn't orbit the sun, the center of mass of the Earth-Moon system does. Earth and Moon in turn orbit this center of mass.
Orbits are a consequence of motion, which is aptly measured by kinetic energy. In order to change an orbit, kinetic energy must be expended in order to slow down, speed up, or redirect the orbiting entity. The orbiting of Earth and Moon is largely conservative, and does not produce nor consume energy, so they cannot alter their orbit.
But wouldn't the acceleration accumulate over a period of time and become noticeable?
No. When you accelerate in your car, you apply acceleration. When you brake, you apply more acceleration (you can prove this by having your friend sit in the car with an accelerometer). The result is not that your car goes really fast, the result is that your car is stationary. Direction of acceleration also matters.
For orbiting bodies, the acceleration is such that it is always orthogonal to velocity, so it only changes the direction of movement, never its speed. It also happens to be quite predictable, so the directional change results in the orbiter going in a circle.
The acceleration imparted on the Earth by the Moon will not make it collide, because the direction of the acceleration is always towards the Moon. Earth already has a large velocity orthogonal to that direction (ie. it is flying "past" the Moon) so the acceleration can only curve its trajectory in a circle.
Generally, you can think of orbits as falling towards an object, but constantly missing.