Okay, I got a new PSU and the problem still exists.
I probably needed to get a new one anyway as my old one was probably about ten years old. Now I have a Seasonic S12II 430W so there shouldn't be an issue with a simple DVD burner not showing it. It's kind of weird because it does show up if I go into the BIOS, but when it switches from the initial bootup BIOS display to the second screen where it shows the final hardware the DVD burner doesn't show up and it doesn't show up in Windows. I also went through and swapped the SATA cables between one of my HDs and the DVD burner. The HD continued to show up in Windows but not the burner.
It's making me wonder if something happened with the DVD burner. It's a new burner that worked fine for a a month or so before it started giving me these problems. I'm not sure how to test a DVD burner as I'm not aware of any utilities to test their fitness the way you test hard drives.