Yes, it is possible for a moon of a planet to move closer to the planet. If a moon moves close enough to the planet, the moon will eventually reach its Roche limit and shatter or else collide with the planet.
Tidal interactions between planets and moons usually push the moons outwards from the planets. A moon which forms in orbit around a planet will orbit the planet in the same direction as the planet rotates, the prograde direction. A moon captured by the planet can orbit the planet in either the prograde direction, or the direction opposite to the rotation of the planet, the retrograde direction.
A moon which orbits a planet in the prograde direction closer than the synchronous orbital distance, and thus circles the planet faster than the planet rotates, will be pulled closer to the planet by tidal interactions.
One example in our solar system is the inner moon of Mars, Phobos.
Tidal deceleration is gradually decreasing the orbital radius of Phobos by approximately two meters every 100 years, and with decreasing orbital radius the likelihood of breakup due to tidal forces increases, estimated in approximately 30–50 million years, with one study's estimate being about 43 million years.
A moon which orbits a planet in the the retrograde direction will be pulled inward by tidal interactions regardless of whether it is below or above the geosynchronous orbital distance.
Many of the outer moons of the giant planets are believed to have been captured by those planets, originally being asteroids or other small solar system bodies. Some of those captured moons have prograde orbits and some have retrograde orbits.
Because those moons have low masses and orbit far from their planets their tidal interactions with their planets are very weak and move them inward or outward very slowly.
Except for Triton, the largest moon of Neptune, which is believed to have originally been a dwarf planet captured by Neptune. It is by far the largest and most massive captured moon, and orbits very close to Neptune in a retrograde orbit. Thus its tidal interactions with Neptune are very strong and pull it inward much faster than any other retrograde moon.
Tidal interactions also cause Triton's orbit, which is already closer to Neptune than the Moon is to Earth, to gradually decay further; predictions are that 3.6 billion years from now, Triton will pass within Neptune's Roche limit. This will result in either a collision with Neptune's atmosphere or the breakup of Triton, forming a new ring system similar to that found around Saturn.
Most massive moons of planets will be formed orbiting the planet in prograde directions above the synchronous orbital distance and so will slowly move away from the planet.
The example of Triton shows that sometimes a planet can capture a large dwarf planet and make it a moon with a retrograde orbit taking it closer to the planet.
The Moon is continuing to slow the Earth's rotation and move outward from the Earth.
But when Earth's rotation is slowed enough to match the Moon's orbital period the process will stop. The Moon will no longer slow the Earth's rotation or be forced outwards.
But the weaker tidal interactions between the Earth and the Sun will slow the Earth's rotation. Thus making the Moon's orbit be below the synchronous orbit and causing the Moon to move closer and closer to the Earth, eventually reaching the Roche limit and breaking up. Of course that process will take tens of billions of years.
This tidal drag makes the rotation of the Earth and the orbital period of the Moon very slowly match. This matching first results in tidally locking the lighter body of the orbital system, as is already the case with the Moon. Theoretically, in 50 billion years, the Earth's rotation will have slowed to the point of matching the Moon's orbital period, causing the Earth to always present the same side to the Moon. However, the Sun will become a red giant, engulfing the Earth-Moon system, long before then.
The Sun should become a red giant in "only" about five billion years. That will probably destroy the Earth and the Moon. If they survive they should come to a standstill in about 50 billion years, and then the process of the Moon spiraling inward toward Earth will begin, no doubt lasting tens of billions of years.
A hypothetical double planet where the two objects have similar masses should hve much stronger tidal effects between the two objects and so they might both become tidally locked to the other in just a few billion years - like Pluto and Charon, for example. And then tidal interactions with their star should start making the two objects more closer and closer to each other.
A red dwarf star would remain on the main sequence for many times longer than the Sun, so it might be possible for a double planet orbiting a red dwarf to move apart, become tidally locked, and then move closer and closer and eventually collide before the star becomes a red giant and swallows them both.
Since the universe is believed to be "only" 13 to 14 billion years old, I doubt whether than has ever happened yet anywhere in the universe. But I will leave it to astrophysicists to calculate the minimum time for such a process.