I am an undergraduate student at a university you have probably heard of who works with their professor on independent research. Recently, I noticed a rather significant technical error in one of the important papers my professor has co-authored. I pointed this out to them in one of our weekly meetings and the professor awkwardly and quickly changed the subject. I am concerned about this but haven't done anything further. I am worried that if I insist on this then it will harm my chances of getting a positive rec letter from my professor down the line when I apply to grad schools. I feel stuck between a rock and a hard place and am very unsure of what to do.
Let me start with a standard question: are you sure that what you noticed is an significant technical error? Have you considered the possibility that your professor's reaction meant something other than a tacit admission that you are right?
OK, let's assume you're right and the paper really does have a significant error. Then according to me:
You have fully discharged your ethical duty by pointing out this error to your professor. In particular, it is not your job to "tell" your professor to fix it nor to "insist" that they do so.
Pragmatically speaking, if you do press the issue with your professor, it is possible they will write a less favourable letter for you. Needless to mention, I don't say the professor would be right to act in this way, but it would be naive to think it couldn't happen.
One commenter recommends that you discuss this with your professor's "supervisor", presumably meaning a head of department or dean. I would not recommend this: if the supervisor brings the issue to your professor, it would not take a great leap of insight to figure out your involvement, and the professor's reaction in this case is likely to be rather more negative in the previous scenario.
In short: you have made your professor aware of the problem. It is now their responsibility to see that it is resolved.