Is the moon visible from every point on the earth?
Yes, depending on the terrestrial latitude, the moon’s synodic position (i.e., with respect to the sun) and the moon’s position on its orbit. That is, there is no place on earth from which the moon is not visible at least some of the time.
Think of it this way: the sun is visible at least some of the time from every point on the earth, right? And the moon’s orbit is quite close to (what we perceive as) the sun’s orbit, albeit a bit inclined to it. So there’s no reason why the moon would be permanently invisible anywhere the sun isn’t.
Well, it revolves around the Earth with a period of about 27.3 days, so everyone gets a look at it. Nearly half the lunar surface, though, never gets a peek at Earth.
Actually, there are lots of points on Earth where the Moon is never visible. Probably hundreds of square miles of area all told.
Consider being down inside a canyon or standing at the base of a bluff or cliff, etc.
The furthest Pole-ward the moon ever gets is ~28 degrees Moon - Wikipedia
So as long as your vantage point obscures all the sky from East to West towards the Equator and below whatever local elevation angle corresponds to 28 degrees above the Equator, the Moon will never appear in the part of the sky you can see.
If we ignore being under an overhanging cliff, that means every place Equator-ward of 28 degrees will have at least an occasional glimpse of the moon. That still leaves ~62 degrees of latitude where the Moon may never be visible depending on where you stand versus the local terrain.
So at 29 degrees North or South you need to be at the bottom of a vertical shaft or cliff to be assured the Moon is always obscured. As you move towards the Pole the required minimum slope of your obscuring wall gets less & less. By the time you get all the way to a Pole, you only need to be in a shallow pit with a 29 degree up-angle to the rim for you to be unable to ever see the Moon.
Also, there are a lot of places inside caves that never see the Moon. Unlike the canyons, cliffs, pits, etc. above, those places may be out of bounds depending on exactly how we parse the OP’s question.
Not to mention all the places under the ocean surface. You might have line-of-sight to the Moon, but it doesn’t meet the OP’s criteria (“visible”) because the light doesn’t penetrate the water.
Heck, if we consider the “surface of the Earth” to be the topmost layer of dirt, not any overlaying water or ice, then 70-80% of the surface can’t see the Moon because it’s under deep water or under a glacier or pack ice.
Even if we ignore the water/ice there are still plenty of valleys, cliffs & pits on the bottom of the oceans to obscure our view just as they do on land. Ditto the mountains of Antarctica.
That just considers places where Nature has rendered the moon invisible. Man-made invisibility is a lot more widespread.
What about inside mines?
Or inside windowless areas in buildings?
Or below decks on ships?
Or aboard submarines?
Those last 4 are toungue-in-cheek, but you see the point.
At the particular lat/long/altitude that is the third stall in the men’s room on the first floor of my office building, the moon has not been visible since the building was built, and will not be visible until the building is demolished. The OP asked about “every point on the earth.” That first floor location has (IMO) more claim to being “a point on the earth” than does the roof directly overhead it several floors up which has a nice view of the Moon.
I love a nitpick as much as anyone, and your canyon/cliff points are great, but I think you took it a bit too far… or perhaps not far enough - what about if you have your eyes shut or if they’ve been sucked out by a vacuum cleaner?
Like I said, I was kidding towards the end.
Here’s an oddball fact for you … the path of the full moon across the sky is the path of the sun across the sky six months later.
This is A Good Thing™, because it means that poleward of the arctic (or antarctic) circle, you get to see at least one of them at all times.
w.
AH-HA! Proof of an intellegent designer 
Except that this factoid concerns only the full moon, and it’s approximately full for only two or three days each cycle.
Still means that the full moon is high in the sky during the polar winters, which is a A Good Thing™.
w.
What if the moon was on a treadmill?
Huge Grin
FML