Well, I've done a little more digging, and I think I've come across a solution.
However, I don't know if this works outside of Windows XP...
First, open up an Explorer window, and go to Tools -> Folder Options -> File Types.
Scroll down to the RPL extension (should be listed as "Toribash Replay").
Now here's where it gets a little weird. I had the same problem as you when I tested in IE, so I changed a few settings until it did.
When you've got the RPL extension highlighted, is the button at the bottom of the window (under "Change") labeled "Advanced" or "Restore"? If it's labeled "Restore", try clicking it then opening a replay in IE.