What happens to a professor if too many students drops out of his or her class? I am taking a course this semester for an upper year undergraduate course in mathematics. 

The course instructor had some experience of teaching lower year courses, and her performance in those courses has been noted as very good. This is her first time teaching an upper year course, her teaching style is minimalistic, instead leaving most of the questions (even the conceptually fundamental ones) to be solved by the students. 

In this semester, she had made this course too difficult. There are many reasons, most of which I think surrounded the fact that she never reinforced any of the material she taught during the lectures and never had any checks in place to assess the class's over all understanding. The homeworks where injected with some tangential, open research type problems, and there were two very difficult midterms that were a bit detached from what she taught during the lecture. 

The course started with 120 students and now we are at about 30 students. We just had the final, which was very lengthy and probably will be harshly graded. We are probably expecting around 25 students to pass, based on class averages from the previous midterms. 

This course does have a steep drop rate from the previous years. But by looking at the course report gathered by some student union from previous years, the course would always start with 120 and drops to something like 70. The lowest was 5 years ago when the course dropped to 50s. 

What are the possible consequences for the professor in this case? She is an assistant professor.
All aspects of university teaching vary so widely across the academic world that it seems impossible to predict how a particular department will respond to a particular teaching issue without much more identifying information.  In this case even the country was not specified.  Usually I can get a pretty good sense of "inside the US" versus "outside the US" but this time I am not even sure of that.

I will speak from my own experience: namely at research universities in the US.  At such a university, being an assistant professor -- which means tenure track faculty unless otherwise qualified, so I will assume this here -- is a long-term proposition.  Although the contract is technically year-to-year, in all of my experience tenure-track faculty members who are not grossly, egregiously misperforming their duties are all but guaranteed to be retained through to the next review period.  Reviews take place roughly every 2-3 years; most commonly there is one such review before the candidate goes up for tenure.  

Because of the above, it is very unlikely that any one teaching experience is going to be the cause for drastic action taken against the faculty member.  For that to occur, the "teaching mistake" would have to be something like deciding not to show up for class for the last month.  Moreover, most assistant professors have a lot to learn about university teaching: even if they have teaching experiences under their belt elsewhere, these will in some ways prepare them for teaching at their present university and in some ways lead them into locally ineffective practices.  In my field (mathematics), it is also pretty standard that most faculty members did their junior teaching at institutions with stronger undergraduate students than their tenure-track institution.  If the crowd is sympathetic, a poor teaching performance early on can actually work out quite well, because the faculty member will be able to show significant improvement in their teaching over the years they spend as an assistant professor: a trajectory which starts out a little shaky and corrects to average could be viewed as more encouraging by many than a flatly average trajectory.  (If the crowd is unsympathetic, you'll be nervous no matter what you do.)

All of this is assuming that the teaching performance in this one class was somewhat poor.  From your description it is reasonable, but not guaranteed, that the professor's performance will be viewed in that way.  The number of students who drop a class varies considerably from year to year -- for me, even teaching the same freshman calculus class at the same institution in different semesters I have had more than 30% attrition (which left me with very good evaluations from the remaining students, so there is something of a tradeoff here) and also less than 10% attrition.  However if one instructor's pass rate is half or less than another's, then that will probably be noticed.  The reaction may not be a punitive one, but the department may work to push this faculty member back to the mean.  (I don't even necessarily agree that this will make the faculty member less likely to teach the same course again.  That really depends on a lot of factors.)

Finally, one small point: you claim that the professor was very successful in teaching lower level courses, is teaching an upper level course for the first time, and has an overly "minimalistic" teaching style, not providing enough help and context to her students while demanding (you say) too much from them. That seems slightly odd to me: in my experience, it would be virtually impossible to be very successful teaching a lower level math class with the style described in the previous sentence.  Maybe the professor thought she had to encourage the more advanced students to be more independent and overcompensated in that direction...but in my experience, that's relatively unlikely.  So I have a little trouble picturing the situation.  Oh, well.