In the most basic strictest answering of your question: Yes it is possible to have a genetic anomaly that causes a larger organ than normal. However, at the levels you are talking it would likely kill the human before they had a chance to pass on their genetics to the next generation. This pattern would occur with birds and reptiles too since the problems that arise have little to do with the species, but with reality being unforgiving.
For genetics survival of the fittest is an important theme. The increased lungs and heart size cause enough health issues that any possible benefit they would provide is negated, and thus their genes would not likely propagate out to the point where all the mountaineers would have them.
Continue reading if you want to see all the challenges they would face...
Assuming normal sized organs at birth
Let's assume that when they are born they have normal lung size. The reason for this is because the human body is like a jigsaw puzzle with how everything fits into the chest cavity. There is no room to expand the heart or lungs down, instead they would have to push the chest out further. With the chest cavity pushed out further it would likely make child birth harder.
Nose and mouth challenge
You increased the size of the lungs but did not increase the size of the mouth or the nose. As a result they would need to inhale and exhale twice as fast for normal breathing. So what happens when they need to do some form of exertion? The result would be even faster breathing, which would put strain on their nasal passageways and diaphragm.
Well endowed women can speak to the kinds of back problems they experience, and your mountaineers would have the same problem. With their chest sticking out twice as much it puts more strain on the spine.
Heart and fluid mechanics
Twice as large heart means it can pump twice as much blood (not really but lets just assume that it just works). It is nice to see the heart is pumping more blood, problem is that the arteries are still the same size. This results in either the heart pumping less frequently or pumping with more force to push the blood through the arteries. In the case of the less frequent your mountaineers would lose any benefit of having the enlarged heart, and thus a mountaineer with a normal heart would have equal suitability of not better. In the later case more force means higher blood pressure which has a whole set of health issues associated with it.
Other heart challenges
Like I hinted at earlier enlarged hearts can have trouble pumping blood and thus not achieve their intended goal. Hearts are designed to work at a particular size, doubling its size causes the design to fail. So having a larger heart can actually reduce the amount of blood it is effectively pumping since it is doing such a poor job at it.
Unfortunately Google kept turning up information on Hyperinflated lungs, which is not the same thing you are doing, making it hard to research the health challenges of having larger than normal lungs. If they are hyperinflated then you will get all the health issues that go with them, if they are larger because they have more bronchioles in them then it might not be too bad. The increased bronchioles would mean more passageway in your lungs connecting them, but the passageway leaving your lungs would be the same size. This would put greater strain on those passageways and your lungs in general.
With the chest cavity pushed out further the ribs would have to be longer to properly wrap around, but they would stay the same thickness. This would put more strain on them, increasing the risk of broken ribs. Considering that coughing hard enough can break a rib, and your people are constantly breathing heavily with more fragile ribs, that even a normal cough for them would have a chance to break a rib.