Q:

How do I address false propositions defended by the lecturer? 

Since I began my master studies (graduate school) in economics I have encountered too many false mathematical propositions.  When I encounter a mathematical proposition which I think is false, I write down an argument (usually a counterexample or a refutation of a premise) explaining why the proposition is indeed false.  Since I hold an MSc in mathematics, I am almost always able to do this. 
Afterward, I present my argument to the lecturer (who may be a professor) who presented, defended and/or used the proposition.  If it is my first time talking with him or her (sometimes I send an email), he or she will usually assume that I am wrong even though he or she did not understand my argument.  The teacher will defend the proposition again and I will again explain why I think it is false.  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.  If I am a bit unsure after I have talked with the lecturer, I almost always read the book (the relevant chapters) or the research articles.  As of now, my readings have just cemented my beliefs in my original arguments. 
So, during the course of my studies, I have encountered false propositions and my lecturers do not seem to respond to critique.  In fact, I question their competency when it comes to discussing mathematics.  In addition to this, they seldom trust my knowledge.  Thus, I have the following questions. 

How am I to pursue my mission to convince the teacher that what he or she is defending is false? 
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? 

After reading Paul Romer's article on "mathiness" among economic theorists, I feel that this problem may be larger than I thought. 

EDIT: Since this is not a forum about mathematics, I thought I would not comment on the false propositions made.  But for those who understand mathematics and mathematical statistics, this is not about propositions which are invalid only if a certain variable is negative or different from zero; because the chain rule was falsely applied; because the hazard function was wrongly characterized; because a limit computation was false; because the conditions for when a matrix is invertible is not satisfied; because the lecturer did not check the second order conditions in an optimization problem; because economists considers infinitesimals etc.  These cases have all occurred.  But other and more significant false propositions have occurred in my program, which has to do with e.g. the difference between applying ordinary least squares (OLS) and applying generalized least squares (GLS) when estimating certain parameters; the consequences of omitted variables in econometrics and under which conditions we can get unbiased estimates by controlling for the correct control variables; the properties of the Nash product; how different assumptions of an error term affects macroeconomic conclusions; the difference between homogeneity and homotheticity of a production function and the consequences this has for certain relevant propositions in mathematical economics etc. 

A:

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 [1].  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...
Footnote:
[1] Based on the author's interactions (mostly with Indians). 

A:

A common defect of mathematicians is to believe that all of science works like mathematics.  In mathematics a statement has a set of assumptions and a conclusion, and the statement is false if you can find one counterexample.  In all other sciences you have some basic knowledge about the world as additional assumptions, and a statement is false if there is a reasonable counterexample.  This is even true in physics, where people integrate the delta "function" from 0 to infinity or believe that there is something like a Bayesian probability.  
To scold a scientist for incorrect use of mathematics is not only petty, but also displays a lack of understanding of science.  A theoretic physicist or economist is not a poor mathematician, but has an insight which allows her to make mathematical "mistakes" while being right.  So attacking a single proposition does not make sense, unless you can invalidate the whole application of the proposition together with the intuition leading to the application of this proposition.