I was experimenting with this sort of thing recently. Try using an "EMBED" tag as follows:

HTML Code:
<HTML>
<HEAD>
<TITLE>Play MyTune</TITLE>
</HEAD>
<BODY>
<EMBED src="MyTune.mp3" border="0" width="150" height="150">
</BODY>
</HTML>
Create a HTML file and put an mp3 called "MyTune.mp3" in the same folder.

I have only tried this on my local machine. I haven't deployed it to a remote server so I don't know whether the file is actually streamed or downloaded.

As CornedBee suggested, any sound that goes through your sound card can be recorded.

Let me know if you have any success with this.

P.S. I couldn't get this to work in FireFox - only IE.