What to do with wrong results in talks?

I either believe this other paper is wrong, or that it is a trivial consequence of my own result

Let me be blunt, if it's wrong and a trivial consequence of your own result, then your result is wrong too. Make up your mind: which is it? Wrong and trivial are not synonymous. The answer differs depending on which applies.

If you believe that the result is wrong.

• If you want to publicly humiliate the other researcher, then you can say during your talk that the other researcher's result is wrong. You will not make any friends and will not impress anyone by doing this.
• Otherwise, your first course of action is to talk about it with the other researcher in private. Have you even considered that you misunderstood something? Most people are reasonable, and if the result is truly wrong, then you will know how to act based on your discussion. Otherwise, this is a whole can of worms, and the answer depends on a ton of factors.
• If for some reason you have not had time to talk with the other researcher, then I would not mention the result at all, and elude the question if asked about it during the talk. If you really don't want to elude it, say that you are aware of a possible contradiction between your paper and the other one and that you are looking into it. (See, that's called tact.) I would also take the opportunity of having the other researcher in the same room to talk about it after the talk, in private.

If you believe the result is a "trivial" consequence of your work.

• If you want to insult the other researcher in public, then say that the other word is a "trivial consequence" of your own work during your talk. You will not make any friends and will not impress anyone by doing this.

• Or you can simply not mention it at all. If asked, you can say that it's nice to see other people being interested in your own work and finding applications.

First, you should know that since your results appeared first (as you stated in a comment), you don’t actually have a strong self-interest to try to make the follow-up results of the other researcher look bad. In a typical scenario, their publishing something that improves or generalizes your result will actually make your own work look better, not worse, including possibly if their improvement is not trivial (actually a trivial improvement may be slightly more embarrassing to you, for example if they discover that a trivial modification to one of your proofs ends up proving a much stronger result. But even then you’ll get the credit of being the first person who published an innovative new proof technique that was used to prove that strong result). So all things considered, it’s pretty likely that your incentives are actually aligned with those of the other researcher, and being as kind and charitable to them as possible (without compromising your integrity, of course) will be beneficial not only to them but also to you. And remember you don’t have to discuss every private thought you have: you may think the improvement is trivial, but at the end of the day, who are you to say? That’s just an opinion anyway. Consider not mentioning it and letting people draw their own conclusions.

What I wrote above pertained to the scenario of a correct improvement. Coming to the other possibility you mentioned, if you believe the work of the other researcher is wrong, then it’s wrong and you shouldn’t pretend otherwise or feign ignorance but simply be matter-of-fact about it - if you discuss the result, state that you believe it’s wrong along with your level of certainty that that’s the case, without gloating or schadenfreude. Don’t say you haven’t had the time to look at it if that’s not true. But you may consider simply not mentioning the result at all if its relevance to what you are discussing is not high.

Incidentally, not mentioning the result may also end up offending the author if they are attending your talk... it’s a tricky business, academia! ;-) Anyway, good luck.

I lean towards "ignore it".

In addition to the issues of academic politics, there is the much bigger issue of confusing the audience. They have limited ability to process information (and math is particularly hard, even for mathematicians). If you mix in a "he said, she said" into your topic along with new results, you end up overloading the presentation. They will have a hard enough time learning from and following the talk, let alone doing justice to an issue of controversy. You just don't have time enough for a good exposition of the rival's arguments and your replies. If someone brings it up in Q&A, deal with it then.

Of course in a paper, you can/should be more all encompassing. But the goals of a paper are different than the goals of a talk.