If you're just talking music, there has been some success with some games. Well, OK, only 3 that I know of. Ultima 3-5 have "upgrade" patches that add in MIDI music taken from other system (C64,Amiga,Apple II,) and converted. The sound effects through the PC-Speaker remained though. You might look into those, and how they did it. You can get the Ultima upgrades Here on the Ultima Reconstruction web site. They also have links to the original sites that has a little detail on how they did it. As for the PC Speaker sounds, about the only way I could see something being done about that would be through a TSR that intercepts the PC-Speaker calls and re-directs them. The quality really wouldn't change, they'd just be coming from your sound system instead of the PC box. I'm not enough of a programmer to even start figuring out how to do that though. Someone might be able to take the PC Speaker code from DOSBox (it is open source,) and convert for use as a PC-Speaker emulator TSR in native DOS. I just don't know. If that can be done, maybe the code for the specialized systems (Tandy, PCjr,) could be used as well.
Not much of a programmer, so someone who is would probably have to:
1: Convert the windows audio calls to DOS audio calls - drivers for different sound cards, or just stay basic SB-8bit computable?
2: Re-route the interrupt or memory calls that usually go to the speaker to the TSR instead.
3: Could probably use the same code that converts the speaker to digital audio though.
4: Wrap it all into a TSR shell.
About the only part of that I would be capable of is the first one. The second one may not even be possible. I know the 4th one is possible (or there wouldn't be drivers for anything,) but don't know the first thing about actually doing it.