Posting so others can learn from my mistakes. I'm pretty bummed and embarrassed about it, but hopefully it'll save someone trouble later.
I had my Hakko 808 desoldering gun at about 700° and I was having a really hard time getting the U2 IC out of a Bally SDU-100. I couldn't get all of the solder off of the component side. I added solder back and tried again to no avail. Ended up turning up the heat and trying to leave the gun on the pads a little longer to try and help more heat transfer to that other side to get the solder to get sucked through.
Ultimately, I lost a couple pads on the solder side and pulled a trace on the component side and I'm not sure where to go from here. Kills me because I was on the verge of fixing my first board from it being completely shot and I pulled this crap.
Moral of the story: don't rush, and stop if something isn't working instead of trying to force it to happen.