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Suppose there is a society where there is a majority consensus on certain principles within science. On laws and mechanisms. However, suppose that minority actually does alternative science that contradicts the mainstream science. This science does not get published by major publishers. Simple becase it contradicts mainstream science.

However, here I assume an idea of open sourse science. Everyone can participate, everyone can read ideas and see mathematics involved. So, it quite soon becomes a place where non-mainstream scientists start to work and cooperate. Instead of merely saying others are wrong just because it contradicts the science they only look for better and simpler explanations. Indeed, every experiment can be explained from different, inconsistent with each other, positions.

But then it appears these theories which are going against the mainstream science allowed to overcome some problems. So, they allowed to advance technology further than the mainstream science. But in this case is it still possible for these theories to be ignored? Given they are open source and more advanced technologies are not hidden.

I consider the society described to be similar to ours. Being heterogeneous in its principles exactly as humanity on the Earth.

As example one might assume a Theory of Everything rejecting relativity but still not being experimentally refuted.

You may be very interested in the work of Thomas Kuhn. He is a philosopher from the mid 20th century who explored precisely the concept you are looking at.

He argued that science follows a sort of pendulum like behavior. It has period where scientists refine the status quo, making small incremental updates, punctuated by periods of rapid change where existing models are swept away and new models come in.

Consider relativity as an example. For hundreds of years, Newtonian Physics was thought to be the "correct" model. Then Maxwell's equations predicted some funny things that pointed towards the need for an "Ether." The Ether was needed to maintain the status quo. Well, experiments drew doubts as to the existence of the ether, and within a relatively short period of time we developed Lorentz Transforms, then relativity itself. In the process, we gave up on precious concepts such as "simultaneity." (relativity does not have a concept of simultaneous events in a global sense. Whether two events are simultaneous or not is dependent on the observer)

In theory, non-mainsteam ideas do get tested, especially if they imply something profound. However, in practice they often go untested because it's just not worth the time and money to go test it. Very few scientists go around disproving perpetual motion machines.

Now you describe two very different things in your question. The first is results which contradict mainstream science. In this case, there is at least one experiment which can be done which is predicted to yield different results depending on if the mainstream is right, or the new idea is right. Eventually this does get tested, especially in your "open science" world.

The second thing you describe is a different viewpoint which provides a different explanation for results, even though it predicts precisely the same results. Science actually does not feel the need to concern itself with this. If theories predict the same behavior, they are "equal." This, for instance, is a major challenge for string theorists. The predictions they make which differ from the standard model require experiments well beyond our current capabilities. As far as we can tell, they are equally effective at modeling the world around us.

Another example are the quantum mechanics interpretations. The fundamental wave equations in QM are agreed upon by virtually all scientists. However, there are very different interpretations of how these wave equations should be interpreted in real life. The Copenhagen interpretation presumes a randomness that comes from collapsing waveforms. The Many Worlds Interpretation declares all observations depend not only on the observed, but also on the observer. Pilot wave theory assumes that it is possible for there to be a constantly varying wave function whose behavior depends on the position of every particle in the universe at that very moment (even those outside of your light cone)

These theories are philosophically different down to their very core. They fundamentally disagree on the nature of our universe. However, they all predict precisely the same experimental results -- results defined by the quantum wavefunction. Because they predict precisely the same results, they are permitted to coexist. They are not called "theories." Instead they are called "interpretations."

For a more current example, consider the work on the use of octonions to form a Theory of Everything. In the article, they mention that octonions are considered to be an underdog in the world of subatomic physics, but that many people find the theory intriguing.

So as for your world? We're in it.

The thrust of your question is #1 very political, and #2 completely ignores what the scientific method is.

Non-mainstream so-called science is not mainstream because:

1. 
   it ignores all the existing knowledge which has been rigorously obtained, and thus constrains all new hypotheses,


2. does not perform rigorous studies with large populations,


3. does not try and criticize itself, and


4. does not make falsifiable predictions.

Any researchers outside the mainstream who do follow the dictates of the scientific method and the four dicta listed above will be brought into the scientific method sooner than later.

Cranks and con-men will rightly stay outside the mainstream of science because they don't follow the four dicta.

It is not possible for accurate predictions to be ignored, regardless of the source. Accurate predictions make money.

If the alt science folks claim to have antigravity and present persuasive evidence, their alt science will quickly be tested and if true, taken over by mainstream science. Unless they are savvy businesspeople, the alt folks with their funny beards will be left behind.

Example: faith healer develops water diet to cure children of epilepsy. Flaky-ass water diet actually seems to be curing children of epilepsy. The Mayo Clinic, perceiving the success but not believing the alternative philosophic grounding of it, evaluate the diet and figure out it is ketone bodies produced during starvation which is the anti seizure principle. They develop the ketogenic diet for childhood epilepsy.
http://www.nevadaosteopathic.org/attachments/article/33/Lin%20Ketogenic%20Diet.pdf

Mainstream science has money and power at its disposal because mainstream science is a tool for those with money and power. The point of mainstream science is to make accurate models and predictions about the physical world. Accurate predictions make money. If my alternative theory is making accurate predictions, people interested more in money than in purity of theory will quickly evaluate and adopt the piece of the alternative theory which produces the accurate predictions.

It depends on your definitions of "ignored," and "successful," I think. There is plenty of pseudoscience still floating around because it makes money, and plenty of perfectly sound science being brushed off because it doesn't make as much money.

First off, technologies could all be open source, but adequate means of production only belong to a handful of people. Just because there is some grand idea or invention that could revolutionize the world doesn't mean it's going to become inexpensive enough for anyone but the most powerful people to produce it. And if said invention unseats the power structure of these powerful people, why would they invest in it? Knowledge is power, and science is knowledge.

Take renewable energy, for instance. Plenty of countries (I'm reminded of Germany and Iceland, specifically) have found that combinations of geothermal, solar, wind, and other alternative power sources can be practical. Scientifically speaking, renewable energy sources are usually agreed to be better both for the world at large, and also for the company selling said energy (renewable energy means a more sustainable business model, hypothetically). However, because renewable energy sources unseat the oil monopoly, they have yet to see widespread international adoption. No one is going to invest in alternative energy sources if they're already making bank on the current energy infrastructure, and no one has the means to invest in said alternative energy except the people making bank. I once heard that you could power most of the U.S if you lined the west coast with power-generating buoys (I have no reference for that, though, so take that with a heaping grain of salt), but nobody's done that. Just because it's scientifically sound doesn't mean it will be adopted, or even accepted by the mainstream, even in the scientific world. I think it's fair to say there's a phenomenon where, if people see something hasn't been done, they will just assume it can't be.