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If a professor is teaching a course with a significant number of students.
My university randomly assigns teaching assistants (TAs) to courses. For example, I was given the assistantship of an advanced hardware course when I was only a freshman.
The TA assigned to the mentioned course has no knowledge about the course topic. Thus, he is making a lot of mistakes here and there.
What happens if a student complains to the professor about the TA doing a poor job?
A somewhat less cynical, but hopefully still realistic take.
First, there is a certain "activation energy" for the professor (and even more so, the department/faculty) to pay attention. The reality is that there is a certain baseline of largely-unwarranted complaints about all TAs, professors, etc., since there is a certain baseline of students who complain about everything, or fundamentally don't get (yet?) that at University, they are responsible for their own learning, and won't get spoon-fed. Of course not every situation/complaint fits that mold, but it has to break out of that noise to get acted upon. Complaints that come from articulate students, with generally good performance in the course and in their academic careers to date, that provide concrete evidence, and avoid name-calling, are less likely to be ignored. And multiple such complaints, not necessarily coordinated, help.
Second, immediate remedial action in the current course depends on the professor. And professors vary widely in their interest in teaching and in their willingness to expend additional time. When I used to teach courses, I would try to attend my TAs' tutorial sessions once in a while, but there were certainly trimesters I never got around to it. However, even then, I think if potential issues had bubbled up to me, I would have made the time. I have subbed in and been "acting TA" for my own course when a TA fell ill during the term, and I'd like to think I would have done the same thing if a TA were hopelessly bad.
Professors generally have office hours. If TAs are good, very often the professor's office hours are poorly attended. A student or group of students could exert pressure, deliberately or not, on a middle-of-distribution-in-terms-of-caring professor by showing up regularly at office hours with questions, and being factual (rather than just complaining) about useful help not being received from the TA. If a professor is regularly being asked "elementary" questions, and hearing the TA could not or would not answer them, it doesn't take a rara avis teaching-prioritizing professor to realize they need to do Something.
In all the above, the unfortunate reality is that it is very difficult for a department to replace a TA during a course. So immediate remedial action is essentially limited to the professor or some other teaching coordinator coaching the TA, and/or the professor doing more of the work themselves.
As to consequences for future courses, again it depends. I wish it were better, but the number of universities/departments that have meaningful performance management and professional development for TAs, in the way I have gotten used to in the industry part of my life, is pretty darn small. The reality therefore is just hope that some coaching helps, and/or there is enough of a clear situation that the TA is not assigned to that course next time.
As to removing bad TAs from the future pool altogether, it depends on the supply/demand balance of TAs in that department. I've taught in applied science-type departments where TA-ship is a building block of financial support for all grad students and it would take someone really bad to be taken off teaching--there's a perpetual hope they'll do better next time around. I've been in a math department where teaching was core, but there were more students than teaching spots and poor quality did have consequences. And I've seen language departments where TA/lector spots were highly competitive and a poorly-student-rated lector was just not going to get hired again.
Finally, decades ago it was pretty awful, but Universities are getting better at policing misconduct. So all the above applies to poor quality teaching by TAs. If the issue is blatantly unfair grading, unresolved material grading errors, or--even worse--misconduct towards students (or others), there are usually now independent processes to take care of that. There may well be challenges whether those processes move swiftly enough and are impartial enough, but it's a whole different ball game.
This answer has a North American, science/math/STEM bias.
My university randomly assigns TAs to courses.
Then it’s not the TAs who are incompetent, it’s the person or people in charge of making the assignments.
Treat the core of the problem, not the symptom. If the system is rotten at its core, complaining about what happens at the end of a broken process won’t result in any meaningful change.
My answer is going to be cynical but it is from a lot of experience. I went to grad school at an R1 public university.
Often the professor will do nothing. The department will continue to assign the terrible TA to the same class, or reward the bad TA by given them an easier (and more desirable/competitive) assignment next quarter. In fact it got so bad that some people were bad on purpose so they could get easier assignments, spend less time teaching and more time on research. If your university doesn't care about teaching, everyone will take the path of least resistance around your complaint.
The one time we got a TA reassigned (he was grading a grad class and deliberately lowering grades of people he saw as competition without being able to explain why) we all pushed back as a group and signed a letter, with some faculty on our side guiding us. He never TAd for us again, but the next year he graded the same class for the younger cohort. The faculty chalked it up to personality differences rather than personal and systemic incompetence.
So your best chance (if an initial complaint is ignored) is to organize, but don't get your hopes up.
What do professors do about TA complaints?
In your scenario, get exasperated by the university. You've explained that TAs with limited or no experience are assigned to courses by the university (rather than the professor or their department). This will frustrate professors, but a system change is necessary to fix, which is likely beyond the professor's ability to influence, especially in the short-term. Longer-term professors can petition their department to push universities for change.
When I was a TA, the professor in charge set up an email filter to automatically forward all course related emails to me.
Someone thought I was incompetent and emailed the professor to complain. The email was auto-forwarded to me. I forwarded it back to the professor because he should deal with it. He pushed it back to me because he did not want to do any work. I met with the student and resolved the complaint.
I have had professors not care, I once TAed a course and gave everyone As so nobody complained about me. Rubber stamp that shit and get on with your day.
In some countries, TA are dispatched after the Professors made their choice on which course to give. As some PhD grants/contracts imply to teach, even in another speciality than what the thesis is on : you might have a TA that never had the chance to choose the course he/she is teaching.
In my case, I was "sure" to be able to teach a course... and when I saw the content of the course : no, I was not ready at all/I had another point of view on the course. And I had to call another TA to correct me at the end of nearly each course (or at least half of the semester). Hopefully, I can teach it now as I learned in parallel with my students.
So, there are tons of good or bad reasons for a TA to not be ready to teach a course. The best things to do, from a student point of view, is to ask questions to the main Professor in charge and stay kind with the TA as long as he/she is not doing wrong things (bad grades, misconduct, ... as stated by Houska).
