I've only purchased a few remasters of the games I own, DMC, FF10, God of War, and Shadow of the Colossus. I also own Jak and Daxter but I don't own the original games, and imo, there's no reason to keep your old copies if you own a remaster, except for one reason which I'll explain below. If you want to play the remaster as it was on the original console, 4:3, then you can do so on either an HDTV or an SD. Change your console to 480p only, and your HDTV to, I think, full if you're looking to play in 4:3.

If the remaster changes the gameplay at all is the reason you want to keep the original. Shadow of the Colossus for example. The exact same game, runs at a better framerate on the PS3, but I still think the PS2 version is better. On the PS3 version there's added frames into the actions of the player. The issue with this is that the colossus will shake a certain period of time and because of the added frames, you can get stuck in a loop on the PS3 version where you go from full stamina to none without being able to hit the colossus once, despite being on their weak point that entire time. The original game you don't have that issue.