Based on some suggestions elsewhere, I tried playing Red Planet & The Neverending Story again.
Since there are no other audio choices other than alternate languages, I played Red Planet in French, German, and Italian, and the offending sound interestingly went away. I also played The Neverending Story in a couple of other languages, and again, the offending sound went away. Going back to English unfortunately brought the offending sound back again.
I then set my audio out to PCM, but the offending sound was still there. I went in a layer deeper and changed everything from Bitstream to PCM, but the sound was still there. I also played with a couple of other audio functions, but nothing got rid of the offending sound in English DTS 5.1.
Anyway, I am getting a replacement Red Planet Blu-Ray from Amazon, so I am going to give that a shot when I receive it. Beyond that, it is still a mystery to me as to what is specifically causing this odd audio problem.