After further research, it's the website where the books are hosted that apparently is causing a problem, rather than the software on the phone.
To dig a little further into the problem, I got a new Motorola Atrix with Android 2.3.4 installed. Using Aldiko with the standard Android browser, I'm able to visit a site like epubBooks and download, save, and open one of their free books. I then had the school upload that same file to the school's website, and when I try to open it using the phone, the file shows up as text.
I did try Opera, and it wouldn't connect the file as an ePUB either (Opera thinks the file is a ZIP file, which technically it is, even though it has the epub extension).
Anyone have an idea about what Android wants in an eBook website that iPhones don't care about?