The Moon is orbiting the Earth in a circular motion. To keep any object in a circular motion we need energy. Hence, does the Earth lose energy by pulling the Moon? In that case, does the Earth's gravity reduce constantly? If it does, then as gravity is proportional to mass, does the Earth lose mass as well?
You are wrong that "to keep any object in circular motion requires energy"
In a circular orbit, the force of gravity is always perpendicular to the motion of the moon, so no work is done by gravity, and no energy is required.
This is analogous to the string in a pendulum, which also does no work.
In an elliptical orbit, energy is constantly converted between potential and kinetic energy, but the total energy is constant, and over the full orbit, no work is done.
So, no, the gravity doesn't run out, and Earth doesn't lose mass.
In a three-body interaction (for example with a comet, the Earth and the sun) the perturbing effect of the sun can cause the Earth to transfer a bit of its kinetic energy to the comet. In this case the Earth can lose (or gain) some energy. The amount of energy lost is minuscule in comparison to the total amount of energy stored in the motion of the Earth. The mass (by E=mc²) is very very small. This process of a three body interaction causing the Earth to do work on a comet is called a "gravitational slingshot". Again the amount of energy lost in these interaction is very very small, and there is not a snowball's chance that the Earth will ever run out of energy.
(Earth does lose a little mass in other ways: a little bit of air is always escaping into space, but there is plenty left, and we won't ever run out.)
Earth does not lose mass to the moon's orbit. However, there are a few ways it can lose energy to gravitational effects. One way it can lose energy is to gravitational slingshotting, from satellites, asteroids or comets. In the case of a gravitational slingshot, Earth doesn't lose mass, it simply slows down. But since Earth is so massive in comparison that slowdown is extremely slight.
The Earth also loses some rotational energy to tidal forces from the moon. In this case, the Earth simply turns very slightly more slowly. Still, no mass is lost.
Apart from the mainly correct replies so far, and just considering the mass of the planet in terms of mass lost to space, or gained from outside, we keep losing some mass from our atmosphere, both from molecules blasted away by solar wind etc, and from the fact that molecules much lighter than N2 (eg water) have typical speeds exceeding Earth's escape velocity.
However, we also are constantly gaining mass from space dust and meteoroids that collide with the planet, thereby becoming meteorites; and I suspect (but have no data) that our input from solar wind plus cosmic "rays" (largely matter particles, such as protons and electrons in terms of mass) exceed our constant leakage into space.
If I am right, then you could say that we still are in a state of accretion as we were early in the history of the solar system, but by now in a very late stage.
In some more billions of years (I have seen various figures) we should find ourselves inside the outskirts of the sun's atmosphere, and accordingly piling in lots more mass, mainly light elements of course. I suspect that when the sun shrinks back to red dwarf status after a few more millions of years, our planet still will be in orbit, and significantly more massive than before, and I would love to see the outcome, but in spite of all my kicking and screaming and writing to congressmen and lawyers, it seems to be very difficult to get ringside tickets.