Is no acceleration a cause or consequence of no net force? If a body is moving with constant velocity, or is at rest, then the net force on it must be $0$. If the net force on a body is $0$, then it must be moving with constant velocity or must be at rest.
Is $0$ net force a consequence of being at rest or moving with constant velocity or is moving at constant velocity or being at rest a consequence of $0$ net force?
 A: Both are true.
The answer could stop there,  but apparently we need 30 characters or more!
A: It is both. Or even indeterminate.
It is important to note that $\Sigma \vec F = m \vec a$ does not express a cause-effect relationship. Causes always preceed effects, so a causal relationship is given by an equation of the form $f(t)=g(t_r)$ where $t_r<t$ or more commonly $f(t)=\int_{-\infty}^{t} g(t_r) \ dt_r$ and $t_r$ is called the retarded time. Therefore, strictly speaking the relationship between force and acceleration in $\Sigma \vec F = m \vec a$ is one of "determined by" rather than "caused by", and it can go either direction.
An example of no acceleration being determined by balanced forces is cruise control where the control system analyzes the acceleration and adjusts the wheel force to match the drag force and keep the vehicle at a steady speed.
An example of balanced force being determined by no acceleration is the normal force from the floor acting on a box. The floor constrains the acceleration to be zero and the normal force is whatever is required.
An example that is indeterminate (to me anyway) is terminal velocity. Both the drag force and the acceleration change until they converge to their terminal values.
A: It is both, depending which object is the subject in the choice of point of view.
A cause is an action that leads to another action.
A consequence is an action that happened due to a previous action.
If you push a ball, the cause is your body moving into the ball and the consequence is the ball moves.
No net force on the ball causes the ball to have no apparent acceleration.
The consequence of your leg missing the ball, when you intended to kick it, is that the ball remains in a state of apparent rest.
Why did I use the word "apparent"? Because if you are in a car, and the ball is in the car, the ball and you will move together at the same rates, and it will look like the ball is not accelerating if you cannot see outside the car windows while the driver accelerates the car gently.
Everything is RELATIVE. We are all moving around the Sun, speeding up and slowing down as our orbit nears the apexes and zeniths since the orbit path is not a perfect circle.
A: Many answers say the answer is “both” or that the question is really philosophical. Perhaps that is true, but it does not seem like a reasonable view to take in this context.
Beginning physics students are usually taught that a force is a “push or pull”. Later, when they learn physics in a slightly more rigorous manner, they should be taught that a force is something that causes acceleration. What other reasonable definition could there be? This is one of the definitions given by Merriam-Webster:

an agency or influence that if applied to a free body results chiefly in an acceleration of the body

It is also the definition given by Wikipedia:

In physics, a force is any interaction that, when unopposed, will change the motion of an object.

This gives us the answer to the question posed here: Having constant velocity is a consequence of zero net force.
(As stated in a comment on the question, at rest is a special case of constant velocity, though whether this extends to moving with constant velocity might not be so clear-cut.)
A: It may be a matter of opinion, but based on Newton’s first law. I would lean towards the latter.
This is Newton’s first law.  It states if a body is at rest or moving at constant speed in a straight line it will remain at rest or continue move in a straight line unless acted upon by a net force. The law begins with the state of motion of a body.
If it said a net force is only considered to exist if there is a change in motion of a body, I might lean towards the former.
The law is due to the inertia of a body which is a property of its mass. I don’t believe the law is based on the property of forces.
Hope this helps.
A: The latter. I think it is most intuitive to think about the F=Ma equation as a statement about cause and effect. The force on an object arises due to something physical (ie. a stretched spring connected to your object), and it's magnitude depends on the configuration of your system. The acceleration is a consequence of the force. Thus zero acceleration would imply no net force, but the reason there is no net force is because he individual forces acting on your objects cancel out.
A: 
Is 0 net force a consequence of being at rest or moving with constant velocity

Yes. It's not a physical consequence, though. Constant velocity (which is also the case for a body at rest) implies zero force (which is connected to acceleration by a factor m), but it doesn't cause it. Force on a body is caused by some agent external to the body. Which is the other way round. Constant velocity implies zero force. Zero force doesn't cause zero velocity. There is no force so how can it cause anything. A finite force can cause a velocity to change.
This is a semantic issue on the meaning of cause, I guess though.
A: This is a question about philosophy not physics. Here is the answer. If I kick a ball it accelerates. It is not because of the ball accelerating that I kick it, for what would be the point of football (EU English) if it was the other way around?
A: The latter.
Every frame is at rest for observers on it. If they note that the objects on the surroundings are passing by at increasing velocities, how they can decide who is really accelerating, their own frame or the objects on the surroundings?
There is a (fictitious) force in the accelerated frame that can be identified by observers on it. Based on that force, they know that they are accelerating.
On the other hand, if there is no net force in the frame, they know that their own frame is inertial. And in this case, if the surroundings seems accelerating, they are really accelerating.
A: Your question is pretty philosophical, but maybe this can help.
Newton's 1st Law, The Law of Inertia basically says that motion continues in a straight line at constant speed unless there is a net force to change that motion.
We usually think of forces as the things that cause motion to change.
In an introductory class, we usually introduce forces with 4 categories of force:
Gravity
Friction/Traction
Compression/Tension
Normal Force/Support Force
When we analyze a scenario, we look for these 4 types of forces.  For the first 3 of these, we generally quote an equation do bring into our statement of Newton's 2nd Law.
Gravity: $F=mg$
Friction/Traction: $F=\mu N$
Compression/Tension: $F=-k \Delta x$
A: As one poster has already said, this is a question for philosophy rather than physics.
Well, according to Aristotle, a force is that which causes change in a material which is capable of changing and does actually change and this by contact.
So here, it is force that is the cause and motion, that is change, that is the effect. We also see if there is no force then there can be no change, that is no motion. This by the way, roughly speaking, is Newtons first law of motion once we recognise that substantial change is not change of position, that is velocity,  but change of velocity, that is acceleration. Of course linearity has to be taken into account too to deduce Newtons first law from Aristotles law of force and change.
A: While the two necessarily go together, an object cannot 'choose' to have no acceleration and thus shield itself from external forces.  However an external agent could 'choose' to apply a force to an object.  The question is borderline physics/philosophy but I think that there's a persuasive cause/effect argument.
A: It is both. When a mass is at rest or in constant velocity there is no force involved. Force has to do with acceleration (slowing down or speeding up) and accelerated masses emit photons.
A: Acceleration causes increase or decrease in velocity, whereas applied force also causes deformation. No acceleration can therefore be understood as the absence of acceleration, whereas no net force may very well be a lot of force. The 'no net' aspect of it merely meaning it doesn't result in any acceleration, whilst it may still result in deformation.
So when it comes to a 'What came first, the chicken or the egg?'-like  question, the answer is clearly force, since force can exist without acceleration, whereas acceleration can not exist without force. Anyone with experience in constipation will be able to explain that to you.
The reason this is still a load of BS is the fact that it matters whether or not get your facts straightened out before you ask a question like this. The beauty in scientific formulas does not lie in the formulas themselves, but in the fact that the image they create perfectly matches reality. You can't just forget a screw and hope it still works, unless you are Roger Penrose and you're busy delivering mathematical evidence for the fact there is no such thing as mathematical evidence. The notion of the force you present is a net result of any (not given) complexity of forces, whereas the notion of acceleration you present does not take into account that, depending on how you look at it, the whole universe is nothing other than a bunch of accelerations. You know, Big Bang and all that. So on the other hand, an acceleration may be a real thing, but a net force is by its definition always an effect of the process of the 'netting' of other forces. It doesn't exist unless you make it exist. Following that line of thinking, its the net force that is the consequence and the acceleration that's the cause.
The interesting thing about this question as is, is in its paradoxicality. We still have to figure out how to shed a light on traveling faster than light in respect to which putting two and two together so far doesn't add up. After looking at things that are natural or real or complex, paradox is the logical next subject of interest.
We know that by looking at things, we change them. What we know very little about is how we change them, depending on how we look at them.
