Does the reaction force appear without delay? According to Newton's law of action-reaction, there is a reaction force for the action force. He did not say when the reaction appears, whether immediately or with a delay.
Could you tell me the truth?
If the reaction appear immediately, does it disobey the relativistic principle that the changes must not be faster than the speed of light.
 A: Warning. I may have misread this question.  I encourage anyone who reads this question to read all of the responses.
No, reaction forces do not "appear" without delay.  If they did, one could use this to send information faster than the speed of light which, as far as humanity is aware, is impossible.
However, when one is doing Newtonian mechanics, one often neglects this fact.  For example, when one does computations involving two bodies orbiting one another, one often assumes as a convenient, good approximation that the gravitational force propagates instantaneously even though it does not: see How fast does gravity propagate?.
A: Brief Summary
In contrast to what joshphysics said (although I understand his point), I would like to say that Newton's third law is instantaneous if you look through a huge magnifying glass!  
Conservation of Momentum (Newton's third law)
The point is, Newton's third law is strictly equivalent (be careful, this is true iff we prohibit action at a distance, see the next section) to the conservation of total linear momentum of a system before and after a collision. This way, Newton's third law becomes as powerful and robust as conservation of linear momentum. And conservation of linear momentum(actually the momentum 4-vector) is something that you put by hand and enforce it inside QFT, and all of its scattering matrix calculations.

Relativity Paradoxes
Then what happens to sending information faster than speed of light and special relativity and killing your grandfather before your father is born and etc...? Well, QFT doesn't have action at a distance! This means all interactions occur in the same space-time point. This will absolutely prevent any of those paradoxes in the first place.
What's the larger picture?
The large picture is, although any of the single interactions (vertices in the above Feynman diagram) preserve conservation of linear momentum instantaneously (and ergo Newton's 3rd law instantaneously), we only see the external particles; or in other words, we never see the intermediate particles. Like, we usually don't see the gravitons carrying the gravitational force between two separated objects! And intermediate particles take time to travel between the interaction points. This means that we will see a delay for Newton's third law's forces (on external particles) to take place. But it's extremely important to remember momentum is preserved for all interactions at any moment (and ergo Newton's 3rd law $\cdots$), unless we can't see the intermediate particles (as we usually don't :-)!
A: The reaction is instantaneous, without delay, and I will show this by a simple thought experiment.
Say you push your hand against a wall. Your hand exerts a force against the wall. Does the wall push back at the same moment? Of course, and here's why.
Suppose there is a delay between when you push the wall, and the wall opposes resistance. Then, during that interval, the wall is pushed with a force, and it doesn't oppose. Strange things happen, your hand goes through the wall for a brief time. Or maybe your hand manages to accelerate the wall with an infinite acceleration, because it doesn't oppose any resistance. But, since we don't see any of these happening, we must conclude that the wall reacts precisely at the same moment.
One respondent said that reaction can't appear instantaneously, because it will allow us sending information faster than the speed of light. There is no relation between these two. Reaction takes place in the same point where action takes place. The wall reacts at the same place where we push it. So, this has nothing to do with faster than light communication. The law of action and reaction is not about transmission of force, but about balance of forces, at the same point and time. It is not about the force at a distance.
In the example of earth and moon, it is true that the gravity exerted by moon to the earth equals that exerted by the earth to the moon, but not because action and reaction at a distance. Consider instead of the moon, an electron which is created together with a positron. It will feel instantaneously the force of the earth, but it will take some time for the earth to feel the force of the electron. One may say that the earth feels the force exerted by the photon which generated the pair electron-positron. Then, what if the photon was heading toward the earth, with the speed of light, of course? Could its gravity reach the earth before the photon? No. So we need to apply action-reaction principle differently. The electron is in the gravitational field, and it is the field which exerts the force to the electron, and the electron's reaction force opposed the field, not the earth directly.

Update. A picture is worth a thousand words.
Action and reaction:
Source.


Not action and reaction:
Source.

Cause and effect:
Source

A: I feel compelled to write a fifth answer because we already have four correct answers, but unfortunately two correct answers say yes, and two correct answers say no. The reason for the ambiguity is that different people are talking about different physical theories. Newton's 3rd law should be construed primarily as a feature of a specific physical theory, which is Newtonian mechanics. Within this theory, reaction forces appear without delay. In a broader context, it is not at all obvious that we can define what Newton's third law means, or that force is even a useful concept. So the short answer is yes, it appears without a delay.

Could you tell me the truth?

Would I lie to you?

If the reaction appear immediately, does it disobey the relativistic principle that the changes must not be faster than the speed of light.

Yes, it does. Newton's third law is a feature of Newtonian mechanics. Newtonian mechanics is a different theory than special relativity.
A: This answer is purely about Newtonian mechanics.
One key point has been mentioned in some comments, but not in the other answers: The terms "action" and "reaction" imply that the "action" somehow causes the "reaction", but this is not true. The "action" and "reaction" are simply a pair of forces. You could just as well label the first force as the "reaction" and the second force as the "action".
There is no way to determine which force should come first, so the forces must appear simultaneously, or in the language of your question, the reaction appears immediately.
A: I am not sure I agree with the above comments.  In the action-reaction principle,  the force exerted by the Earth on the moon equates in magnitude that of  the moon on the Earth full stop.  I don't think there is any 'time delay' per se as implied in the question i.e.  it does not take time to the moon to build up a reaction to the gravitational pulling from the Earth.  That's fundamentally because we are precisely looking at an inter -action and that no body has an active role while another would be passive,  this is just a matter of point of view.  There does exist a delay due to relativity but this delay accounts for the fact that an interaction at time t originates from fields created by the objects in the past and as far as I know these delayed potentials do not introduce any asymmetry actor-reactor .
A: The action-reaction forces of Newton's Third Law always occur simultaneously. This is not exploitable to send information faster than light, it is simply a necessary occurrence.
The action-reaction forces occur simply because momentum is always conserved. A force is defined as a change in momentum over time. An action force would be described as imparting momentum onto something. You increase its momentum with an action force. But because momentum is conserved, it must come from somewhere. The momentum is drained from something else. That would be the reaction force. The action force gives momentum to one thing while the reaction force is taking that same momentum from another thing.
This must occur simultaneously because momentum is conserved. Imagine you have a container separated into two halves by a removable barrier and one half is completely filled with water. Then remove the barrier. To ask whether the reaction force is delayed from the action force is like asking if the full half empties after the empty half fills. No, they happen simultaneously because the water filling the empty half must be simultaneously emptying the full half. Similarly, momentum is conserved, so the action force adding momentum to one system must be simultaneously balanced by a reaction force draining that momentum from another system.
A: One answer says that the reaction is instantaenous by considering the example of the hand pushing against the wall.
It was said that if the reaction force is not instantaneous, the hand may go through the wall for a brief time, which is not allowed.
That answer is wrong. When a hand is pushed against a wall, what really happens is that the atoms on the hand comes close to the atoms on the wall and feels electromagnetic repulsion due to the electromagnetic field of the atoms on the wall. This electromagnetic repulsion is the reaction force and since the electromagnetic force is infinite when two atoms are in contact with each other, the atoms on the hand can never touch the atoms on the wall. This is true even if the reaction force is not instantaneous.
And we know that electromagnetic signals can't travel faster than the speed of light. The atoms on the wall will feel the electromagnetic field of the atoms of hands at a delayed time. They can only respond to the this electromagnetic field at a delayed time. Thus the reaction force cannot be instantaneous.
A: In Newtonian Mechanics when you are talking about action and reaction it is on a rigid body , relativity the concept of rigid body is not valid. the moment you hit a wall (even if you are hitting with perfect plane sheet on a perfect plane wall. Wall undergoes small deformation  and reacts.While your elbow continues to travel with the same old velocity and when the reaction reaches back the elbow it stops! means a deformation and subsequent correction to older state takes place ,same is true to the wall action with finite speed reaches different layers and each layer reacts.since it is fitted to the ground same as your feet and when there is no movement action and reaction cancels each other!        
A: Alright, to truly understand this concept you are baffled about, you must consider the most fundamental causes of the force. Although many different types of forces could be written to be expressed in same unit, they may have fundamentally different reasons for the interaction they create. First consider an interaction between electric fields. When two fields encounter one another to form a different net field, the change in E-field is delivered to the source electron in the speed of light, which is the propagation speed of electric fields. So, in this case there will be a delay in the reaction force.
Now consider yourself pushing against a wall. This case it is physically impossible to have a delay in reaction force. This sounds contradicting as forces seem to act differently compared to aforementioned case. But the key is to notice that the interaction itself happens instantaneously. Specifically, force I create with my fingers originated from mechanics of my body and when the finger pushed against the wall, interaction happens instantaneously, causing the reaction. Similarly, when two electrons interact, the fields deliver the forces to the electrons and take time, but the interaction between the fields itself are instantaneous.
