No, the moon will not leave the Earth. The rate of increase gets less and less over time. Eventually, the distance will stabilize -- except that is so long in the future, the sun will expand and engulf the Earth and the moon together.

Fun fact about gravity: it is the weakest of the 4 fundamental forces but has infinite range!

One theory states that the moon will eventually slowly creep back towards us eventually becoming a ring (like Saturn's rings)... If the Sun doesn't engulf us by then.

>Eventually, Earth’s rotation will slow until it, too, is locked to the moon’s orbital period. At this point, Earth will still be rotating faster than it orbits the sun and the smaller tidal bulge due to the sun’s gravity will continue to slow Earth’s rotation.

>Earth’s tidal bulge will then begin to lag the moon’s orbital motion, the pull will be acting in reverse and the moon will slowly start to spiral back towards the Earth.

>The moon will move ever closer until it reaches 18,470 km (11,470 miles) above the Earth, a point known as the Roche limit. This is the radius inside which the tidal forces pulling objects apart exceed their mutual attraction due to gravity.

That would be theoretically possible. However, the Sun will reach its red supergiant phase and engulf the inner planets long before the Moon could escape.

Which also provides the literal answer to "when will the world end?" When the Sun feels like it, that's when.