Disclaimer: I am primarily a student of physics so I'm not sure if the same issues arise in studying economics as well. Since your main question is broad -- it would be applicable for any discipline with complex mathematics -- I am writing an answer, albeit from a physics perspective.
First let me summarize some of the points made in the comments and add a few of my experiences.
It is plausible that an economist or an engineer would use a heuristic argument to "prove" an statement, even though this is not proved in a mathematically precise way. For instance, the "proof" may not work everytime, but it may work often enough. Cases where it fails may not be of interest or not easily characterizable, which is pretty much the opposite to what we do as mathematicians. I've seen some fields that work like that. So the issue may be a lack of common language between the OP and the lecturer. - Shake-baby
Have you looked into this? One of the most common cases of derivations being unsatisfactory in physics (in my undergrad) is that a LOT of context is often implicit (as opposed to being explicit in math). Frequently, one needs to be aware of several small points (mentioned somewhere way back in the text) while understanding new concepts. Even more frequently, one proceeds to do calculations without explicitly stating important conditions.
Examples of what I think would be at least slightly alarming to mathematicians:
- Functionals will be considered without specifying which function space is the domain. Also, we often ignore measure theory entirely (in formal training) even though we see integrals over function spaces.
- The level of differentiability will almost never be specified. The implicit assumption often is: take as many well-defined continuous derivatives as needed.
- Naked "delta functions" are welcome. The more, the merrier.
- Physical intuition is often more valuable than a solid definition. In a recent graduate class, we were discussing the fundamental group for special cases like a torus and a projective plane by cutting up paper and playing with rubber bands. However, topology was not a prerequisite for this course (!) and several of my peers had not studied it formally (neither did we define "topology" in that class).
If experiment matches theory, all's well and fine. Is there a mismatch directly because of wrong assumptions? Cool! We've got new physics to play with now. Does this hand-wavy approach mean that physics is "wrong"? It simply means that the level of rigor in physics in insufficient for a mathematician but is (often) perfectly fine for model-building or experiments.
Suppose you are at a stage where the former point (of common context) is not an issue and still you are facing major difficulties in communicating.
How am I to pursue my mission to convince the teacher that what he or she is defending is false?
First, I would like to point out that "to convince the teacher that what he or she is defending is false" is your mission, should you choose to accept it. No external agency has thrust this responsibility on you. That said, there might be many good reasons why you might feel that you should do so such as:
You want to help correct a misunderstanding of the professor for his/her benefit and the benefit of other students.
You consider that stating and believing false statements to be true is a bad thing by itself and so you want to correct that. This is different from the first point in that this point would apply even if you were the only student in the class.
Fortunately or unfortunately, convincing people is hard. Convincing people that they are wrong is much harder. Convincing people that they are wrong and that you are right is much, much harder . Sometimes it's easier when the opposite person is a scientist. Sometimes it's harder. You try your best and reason with the person in good faith; it will work sometimes and sometimes you just move on after being unsuccessful.
How can I approach the teacher with respect (I am more like an activist than a politician) and send a message that my knowledge is important and that I am of the opinion that I know something to be false which the teacher think is true?
Permit me to break me down the question bit by bit:
How can I approach the teacher with respect
Be polite in words and manners and act in good faith. Some professors might get annoyed if sigh loudly. Others might not even notice or care if you yawn while they are making a key point. There isn't a silver bullet. The teacher is also a person, just like you. Is the teacher is trying to impeded your learning? No. Remember: the fallacy is the key issue, not the teacher.
(I am more like an activist than a politician)
If I understand correctly, you mean that you take speak in an argumentative manner as opposed to a soft tone. I have sometimes been guilty of this myself, especially when I felt what the opposite person is saying is very wrong and very stupid.
One possible solution is: write, instead of speaking in person. I understand this may not be the most practical in a lecture setting but courses often have online forums which make this more convenient. An email (or forum post) allows you to (i) collect your thoughts, (ii) frame your arguments fully and (iii) most importantly, gives you time between framing your points and actually clicking send -- in this time you can go over the language and double-check it (or have a close friend look over the email) and alter it if needed so that your tone does not come off as hostile.
Consider the following two emails:
(1) On Tuesday, you said that the isomorphism that takes the fundamental group with one base point to the fundamental group with another base point is path-independent. That statement is incorrect. It is true if and only if the fundamental group is abelian.
(2) On Tuesday, you said that the isomorphism that takes the fundamental group with one base point to the fundamental group with another base point is path-independent. Isn't it true only when the fundamental group is abelian? At the time, had we already assumed that the surfaces under consideration had abelian fundamental groups?
Perhaps you think that these two are roughly the same/interchangeable. Perhaps not. I would consider the second one more polite and preferable compared to the first one. Moreover, the second version expresses two things which are missing from the first -- (i) a possibility that you misunderstood something (humility) and (ii) a desire to arrive at the right answer together (cooperation). In contrast, the first version just says, "Here is the right answer. You are wrong." (superiority).
and send a message that my knowledge is important
It is the truth which is important. In this special case, your knowledge coincides with the truth ... but are you always correct? Probably not. If a teacher is dismissing your claim as false, don't take it personally (trust me, it never helps). This can get really hard, especially if it happens publicly, but you must keep your cool.
After a while, he or she will either ignore me, reject that I have something valuable to say, accept my argument, or tell me that I should read a certain book or certain research articles.
There you have it: there are all different kinds of professors. Is that a surprised? Some accept your arguments, some don't ... surely, a reasonable academic would accept an argument if it was true? Unfortunately, life isn't so simple...
 Based on the author's interactions (mostly with Indians).