Is there any friction between wheel and road? Human beings invented the wheel to get rid of the friction between the wheel and the road. But were we able to reduce it to zero? Is there any residual friction?
This question is about only the friction between the wheel and the road. I understand that there will be friction in other places, e.g. between the wheel and axle or between the vehicle and the atmosphere.
 A: As many others point out, there is friction present, otherwise the wheel wouldn't grap the surface and pull the car forward. But you are talking about a different kind of friction. There is a possibility of different kinds of friction:

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*Kinetic friction, if the wheel ever slides and skids over the asphalt. This is friction between objects that slide over one another.

*Static friction, which is what the other answers talk about. This is friction that holds something still. It always works in the direction that prevents two objects from sliding. The point on the wheel that touches the ground experiences static friction, because it is standing still in that very negligibly small moment.

But

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*rolling friction is what you are refering to. Ideally there is no kinetic friction, and static friction only grabs the asphalt and doesn't reduce the speed (on horizontal surface and without wheel torque). All other forces that do work against the wheel rotation (except friction around the axle, as you also point out) are collectively called rolling friction.

Rolling friction happens of several different reasons. For example,

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*the rubber tires contract and expand and thus dissipate energy. The energy is taken from the rotation, and this factor counts as rolling friction. Also

*the ground underneath might deform. The deformation costs energy and will as well cause a surface with normal forces that do not act radially (towards the wheel's center) anymore. Such forces will cause torques that might counteract the rotation. See the following picture from this source:


Without rolling friction (in an ideal world), the car will continue to roll and never stop. I believe this is the actual question that you have. Because you are right that in this sense, friction counteracting the motion has been eliminated as you describe.
A: Without friction between the wheel and the road, you would not be able to accelerate (positive, negative, change of direction). 
So it would be impossible to drive a car with zero friction tires. Except you want to use the wheels only to maintain the current state of movement and you do steering and accelerating by other means. 
Hovercrafts or magnetic levitation trains use "zero friction" but levitation doesn't move or stop them.
A: Friction is the reaction force between two objects as they move over one another such as a car's tyres and the surface which it is travelling on. There is no question of friction being absent. It is always there, though it is very less as compared to sliding friction.
And humans did not "discover" (I prefer discover over invention for various reasons) wheel to get rid of it but to make their work easy.( I don't think humans at that time knew anything about friction or for that matter force)
Another thing, friction being a reaction, not exactly a force is not something one can get rid off. Anyways everything which moves
(even in space : Long range spacecraft's speed and direction  need to be adjusted from time to time or else they would be deviated from their path. Space isn't entirely vacuum.
The fact is that space is filled with tiny particles called cosmic dust and elements like hydrogen and helium. This applies for interstellar space also and all the previously mention particles make up what is known as the interstellar medium.
The interstellar medium is mainly made of lone hydrogen atoms. They do not even exist as pairs as they do on earth. I mentioned before that space is filled with hydrogen atoms. The actual density of hydrogen as it exist in interstellar space is on the average of about 1 atom per cubic centimeter.)
has to overcome friction.
A: Wheels reduce friction because the point where the wheel contacts the road does not move relative to the road. At any particular time, the point of contact is stationary on the road, as the wheel's contact point is moving backwards at the same speed as the car is moving forwards.
I had a great opportunity to check this empirically some years ago. I was driving behind a van that was trailing a good 10m of thin rope on the ground. By moving slightly left to right, I was able to get the front wheel of my car to pass over the rope. Instantly the rope snapped. It did that because the section of the rope between the ground and my tire suddenly was stationary, whereas the rest of the rope was moving at some 60 km/h with the van.
Of course, as Timeless points out, there has to be friction between the wheel and the road. If there wasn't you could not accelerate, decelerate or change direction. To test that, try driving on ice, where the friction is much reduced. 
However, if you are driving at a constant speed with ideal wheels (a good example is railway wheels on a track) the friction is far lower than if you had to drag the car along without wheels. Good tires are designed to give low friction when driving straight ahead at constant speed, but also provide high friction when changing speed or direction.
A: Of course, there is friction. Otherwise the tyres will never wear away. You will change them once at 10-15 years because of material aging.
