What's going to happen if I just use the web interface to try and install OpenWRT on e2500 v4?

Linksys e2500 v4 is now supported by OpenWRT. A very helpful & thorough guide was created by one forum user showing how to install it on the router, including soldering pins together and other such commendable wizardry.

My dumb butt is intensely curious: can I not just install it by selecting the file to upload through the "upgrade firmware" web interface? What kind of hilarity will result if try to do that?

I don't think it would be as hilarious as you expect, I highly suspect the router's WebUI will complain that you're trying to use an incompatible file for upgrade.