I remember removing IE (including the core) with nLite a long time ago (probably 2 years ago), and I remember that whenever I ran an app that used the IE core (like Steam), it would pop up an annoying "Save File As" window whenever it tried to use the embedded IE control.

I'm wondering if this has been fixed since then, since I'm making a nice lean XP install, and I figured I'd take a shot at removing all of IE since I hardly ever use it. So, does it still pop up that annoying box?

Most likely. Steam is designed to work only with IE, not some other browser, so if you remove IE, regardless of whether you use some other browser and have it set as default, applications that depend on IE, like Steam are not going to work. Best bet is to remove IE but leave IE Compatibility.

What is exactly problem here?
You remove IE core, it doesn't recognize the html then, it's like any other file to download, unless you use some third party browser which gets set as default browser.

Yes, but other browsers don't get integrated with windows properly so the programs still mess up; for example in PerfectDisk 7, if I took out IE Core the intro page it uses would always load in a new window in Firefox, instead of in the program as it should. I doubt that has changed, and some other programs depend on it more than for an intro page.