Does a department have any obligation to respond to a student petition against a teacher?
Such a petition is something that a department head or dean would take seriously.  But at most universities, it would be their decision what action to take, if any; a petition wouldn't obligate them to make a change.  

Consider that if such petitions were binding, it would give the students an undue amount of leverage over the professor; if they got a difficult homework assignment or something else they just didn't like, they could threaten to submit a petition to get the professor removed.

There is a principle of "academic freedom" that the administration will generally avoid interfering with how courses are taught, within reason, so it would probably take a pretty egregious deficiency for the administration to take the disruptive step of removing a professor in the  middle of a course.

It is likely to be more effective to start with one or two students having an informal discussion with the department head about the situation; a petition doesn't leave much opportunity for dialogue or compromise.

Note also that the department head's decision would likely be based less on the students' opinion of the course, and more on objective information about what the professor is actually doing.  So it would be more helpful to avoid phrases like "everyone hates this class" and prefer specific examples: "the professor did this and this, and here is why we think this is a problem".