There are really two things here. Firstly, "stressed" yeast. Secondly, mutated yeast.
"Stressed" yeast are purely physiologically in a sub-optimum (for growth) state. They are going to have limited viability. This means that they may not reproduce readily if you put them into a sub-optimum culture medium. If you harvest a "stressed" yeast and try to put it straight into a high gravity, high IBU, low oxygen wort, it might not get to take off. However, if you harvest viable stressed yeast (by yeast washing, for example) and allow them to "relax" by placing them into a nice 1.040 unhopped starter with lots of nutrient and oxygen, they will quickly re-adjust and do just fine. Stressed yeast can return to unstressed yeast, and make you great beer again.
Now, mutations are different. Every culture of yeast is going to harbor genetic variation. With enough reproduction cycles in an uncontrolled setting like a homebrewery, the "wrong" yeast are possibly going to start getting more of a foothold in your particular sub-culture. These yeast are going to give a slightly different malt profile, or have altered alcohol tolerance for example. I have heard that if you keep re-using yeast, it will last about 6 batches of beer before it starts to give subtly different profile to your beer. Some people claim that supposedly the same strain from White labs and Wyeast can have different profiles, possibly due to different culturing like this. Now, in a microbiology lab, the original strain can be kept pure by a variety of methods. But not so in "the wild" as it were.
SO what happens when you combine stress and mutations? Basically, a stressed out yeast is going to be less viable, for the reasons already mentioned. It will have to adjust its metabolism before it can start to grow. All of this gives "other" varieties within the same culture (i.e. you beer) a chance to grow a little bit better; reproduce a little bit more. Yeast which are fitter will have a slight competitive edge, and come to dominate. "Stress" (such as low oxygen, or high alcohol) is a selective pressure. The fittest yeast will grow fastest. The population will adapt, because of genetic drift. Basically, the more stressed a yeast is (the more selective pressure that is applied to it) the faster the population will evolve.
Did you realize that everytime you make beer, you are providing more support to the Theory of Evolution (first published 150 years ago today!)?
Now, the actual changes that take place in your brewery are going to depend on a lot of factors. Where did you get your initial batch of yeast from? How genetically pure was it then? What stresses did it get subjected to? How well did you recover active yeast? How many reproductive cycles has it had (what generation is it)? So to say NEVER use yeast from a 1.060 beer is being to absolutist. You need to judge a lot of other factors too.