How does it affect the treatment of a mathematician's results, if that mathematician was a Nazi?

I think there is no academic issue here.

The problem with Vahlen, as you may have guessed by the title, is that he was about as much of a Nazi as a person could be. He was in the SA, the SS, accused brilliant Jews of plagiarizing Aryans, helped the Third Reich expunge Jews from the scientific community, etc. He even supported the Nazi party before its infamous rise to popularity.

I looked into Theodor Vahlen, and I agree that he was a morally reprehensible individual. But this is irrelevant to a discussion of his work: he proved the theorems that he proved whether he was good, bad, or whatever. As is well known, the Unabomber is a published mathematician. His six papers have eight citations on MathSciNet, three of which came long after his capture, e.g. see here. (Though I do not know for sure that the author of this paper was aware that the T.J. Kaczynski whose work is cited is the Unabomber, it seems likely: the author is an American, and among Americans this name is well known to say the least.)

I was wondering what the likelihood is that his results were neglected due to this political misfortune. I was also wondering if people know of similar instances, particularly in mathematics. It seems like whenever I look up the politics of a respectable mathematician, they are either nonexistent, or equally respectable. I don't know if we tend to discredit people like this, or if people like this tend to not produce good results, or what.

In my opinion this is unlikely. There are very famous mathematicians who were, to lesser or greater extents, participaters in the Nazi movement. Perhaps Ludwig Bieberbach and Oswald Teichmuller are the two leading examples: these are household names in mathematics, and that they were intimately (and reprehensibly) involved in the Nazi movement is also very well known. (For that matter, I am not convinced that Vahlen's work is so little known. It is described in the wikipedia article linked to above, for instance.)

In my personal opinion, it is worth knowing when a mathematician (or other academic) has done reprehensible things in their personal life. Some years ago I compiled a list of "greatest mathematicians" on MathOverflow: I limited myself to choosing at most one mathematician born in a given year, but in some cases I included "honorable mentions." When I got to Teichmuller, I did not want to use the word "honorable" to describe him, so I wrote "dis/honorable mention." That was a personal decision (and by the way, I am partially of Jewish descent, so it is really not for me to forgive or forget such things). But I included him on the list anyway: that he was a terrible person does not influence his mathematics one way or another.

I'd like to just use a big-ol' script V for the Vahlen matrices (which I have not seen in any papers thus far) and move on,

There is no problem with this.

but I don't want even an iota of a suspicion of being sympathetic to this man's priorities.

We don't endorse the life of Teichmuller when we talk about Teichmuller space or Bieberbach when we talk about Bieberbach's Conjecture. We credit them with their work, as we must.

It would also be hilarious to just use a swastika, but that joke might not be funny to some people (understandably).

I am not laughing at all. Please do not do this.

Finally, you can read much more about mathematics in Nazi Germany from this book. Again, I think it is good to know these things.

I was wondering about the etiquette of discussing a result by such a person. Ahlfors simply writes about his math, which seems probably the right way to go. But it still feels weird if I were to give a lecture where I'm praising a guy who may have contributed to the extermination of an audience member's family.

You need to remind yourself why you are there giving a lecture to this audience member in the first place. What is the goal that your lecture is trying to achieve?

Is this a lecture about mathematics?

or

Is it a lecture about the history of Nazism?

If this is a lecture about mathematics, talk about mathematics. Think about it this way: if you invited a speaker to give an algebra seminar at your department and they instead went on a 10 minute digression about the history of automobiles and inserted icons of automobiles to stand for mathematical objects instead of Latin or Greek letters, would you think that was appropriate, or would you think that it detracted from the goal you had in mind when you invited them to give a talk, namely to learn about their work and mathematics related to it?

It's okay to tell little historical anecdotes about the people behind the mathematics in talks; we are humans, not robots. Nonetheless, I'm willing to bet that your hypothetical audience member whose family was exterminated by the Nazis is still there primarily to hear about interesting mathematics and not about historical anecdotes or to see cheap gimmicks ostensibly making fun of the Nazis, and ultimately doesn't care that much to have any of that stuff brought up. Your use of a swastika as a mathematical symbol may or may not offend him; will certainly not strike him as funny; and will probably just annoy him for being a distracting gimmick that undermines the primary goal of the talk.

To summarize, no one really cares what "feels weird" to you - that's your own private feeling; to the extent that that feeling is causing you to want to give an unfocused talk that goes on irrelevant tangents and annoys people, the correct thing to do is to ignore that feeling and instead give the best mathematical talk, which focuses -- as a mathematical talk should -- on the mathematics. (Hmm, and that's basically what Ahlfors is doing in his writing isn't it...?)

As @Rudiger says, we should tell the truth so far as we know it (and make an effort to know it!), and give credit where credit is due, regardless of the possible failings in the other parts of a person's life.

I disagree with @Rudiger's claim that mathematics has nothing to do with politics, at least insofar as all these things are mixed together and affect each other in the larger human enterprise. After all, Nazism affect mathematics profoundly, by affecting mathematicians profoundly (killing some, driving some to suicide, and driving others to the U.S. and other places far away from Germany).

So far as I know, in Vahlen's case, the lack of attention is not so much about his abhorrent politics but that people did not find his mathematical work terribly useful (whether or not you yourself do).