I did some quick searching to make sure that burlap will take dye - which it will, since it is 100% jute, a natural fiber - and then stumbled upon Rit's amazing color guide. Seriously. It's amazing. It has a whole section called "Yellow-Orange-Peach". Perfect!
Intentionally I wanted to a color (#79, if you're curious) out of the "Orange" section. My impatient self wasn't willing to drive all the way to Jo-Ann, so I checked out the Rit dyes at my local Wal-Mart. No cherry red or golden yellow - the only two colors I needed. So I whipped out my phone and checked all the orange-pink colors to see if any of them were a) an acceptable shade, and b) used only colors available to me. Luckily, I found #464, which requires scarlet and sunshine orange, two colors that the Wal-Mart carried. So I grabbed the dye, four yards (yes, four - just to be safe) of natural burlap, and a large plastic bucket.
I can't wait to make the lampshade. Sadly, that project might have to wait until I'm situated in the new house. And I still need to figure out what lamp base I'm going to use...