Can revving the engine in neutral cause damage?

It is indeed mostly a rumor, but with some background.

Temperature

The main problem is with the cooling. If you rev the car while standing still, the radiator will receive less air, thus it will cool the coolant less efficiently. Usually the fan will force some air to the radiator if the car is not moving fast enough, (like if you are stuck in traffic). This means that the engine will operate in the higher threshold of temperature. The problem will be worsened if the car already has cooling problems.

Unnecessary wear

The secondary is that you are turning the engine and naturally wearing it without moving the car. The engine wears for a combination of factors and one of the most strong of them is the number of revolutions it will do over its lifetime. It is like driving the car for more miles that it actually has in the odometer.

Either way, both factors only offset the engine wear for very little, unless if you keep revving the car for 30 minutes then driving it for 5 miles a day. The car will seem to have a low mileage, but the engine will wear out much earlier than other comparable engines.

In my humble opinion, to take an engine to high RPM under load (where the piston is cushioned at the top of the stroke by a large charge of fuel/air) is far better than to reach the same RPM unloaded. An unloaded engine, with a relatively small fuel/air charge, relies entirely on conrod bearings to stop the piston at the top of its stroke. in a "loaded" engine, the large charge and therefore high compression pressure, cushions the piston action.

Depending on the year/model, the rev limiter will not allow you to redline in neutral.It will cut off the engine before the rpms can do damage. Usually cuts it off at 4000 rpm - depending on year/make/model.

Don't listen to any BS people say here... If the engine is in the operating temprature, it is okay to rev it getting not too close to redline. And not revving it for long periods of time... I treat my cars like that and they are still golden.

Yes, this causes engine wear.

When the transmission is in neutral and the engine is “revved” without any load, the spinning engine internals will accelerate, gathering rotational and lateral forces at a faster rate than designed by the manufacturer.

Why will the engine wear? (not an exhaustive list):

Piston compression ring expansion

Rapidly revving an engine will heat up the piston rings much faster. Because they have a much smaller thermal mass than the cylinder liners they expand at different rates. If the compression rings expand too much, they will generate increased friction on the cylinder liners, causing the cylinder liners to wear out (reducing compression).

In the worst case scenario, the compression ring ends touch, pinch the cylinder liner and will most likely cause the piston to crack. Then you’ll have metal fragments flying around destroying the head and cylinder.

On a very cold engine or while lean (not enough fuel), it is easier to get a larger thermal differentiation between the cylinder liner and piston rings.

Some early rev limiters implemented used a fuel only cut which could cause a cylinder to lean out "slowly" and develop a piston hot spot from detonation.

Contrary to other comments & answers given, the cooling system is NOT able to help, as piston failure happens within 4-10 milliseconds after a hotspot develops on a piston or compression ring. (See video for references to fuel injection millisecond times below)

Oil system starvation in wet sumps

More common in continued over revving - the cylinder head & block doesn’t drain the oil at the same rate that oil is pumped out of the sump, leaving the sump empty, common in the Rover V8 engines. This is very common in engines that don't have the minimum oil required. Engines that are not serviced have dirty oil systems and are prone to blockages at high rpm.

This can also be caused if the block uses the same pipes to drain the oil that is also used to ventilate the sump causing oil vaporisation. However the engine would most likely already be overheating for this to occur.

Some poorly designed oil pumps (and water pumps too) can aerate and are unable to pump oil when increased in rpm too rapidly.

The next two points are much more common in performance engines where the manufacturer has tweaked output leaving no room for improvement. You’ll find plenty of videos on YouTube of motorcycles and Italian supercars revving without any load and destroying themselves.

Engines that can handle repeatedly being held on the rev limiter without any failure generally have a conservative rev limit set or have lots of performance parts available.

Connecting rod (conrod) warping

Some conrods will stretch/bend during excessive rpm acceleration. Worst case scenario is it will put greater unbalanced forces on the crankshaft and bearings (higher compression ratio only needs 10 thou). If the engine has tight tolerances it will also bend valves.

Crankshaft warping

The crankshaft or engine block isn’t designed to withstand such a sudden increase in internal force (similar to harmonic imbalances). It only takes a thou or two (0.0254 - 0.0508 millimetres) for the crankshaft to gouge a main bearing. This is common in performance & motorsport engines so they have increased strength built into the block by adding more ribs and webbing to the cast or billet.

If you’re curious what a “catastrophic” harmonic engine failure looks like at 11,000 rpm with load on a dyno, watch https://www.youtube.com/watch?v=1LkxGx5WJzA and skip to 14:20 where Cosworth gently explore the limit of their turbocharged 4 cylinder during F1 engine research & development.