Do replacement DVDs save progress?

Game progress is saved on the Xbox hard drive, not on the game's DVD. So your saves are going to be fine if you can get another disc.

Edit: I tried to google a bit to find some supporting evidence, and man is it hard to find. I mean, there are plenty of sources that will tell you the XBOX 360 has an internal hard drive, but basically nowhere that explicitly says the DVD drive is read-only. I mean, it is, I'm sure. But even xbox.com is letting down in terms of actually saying it.

Your progress will persist if you get a new copy of Arkham Origins, because as a few people in the comments mentioned, the DVD is read-only, and if it weren't, writing and re-writing save data every time the game auto-saves or you do a manual save, would take a long time to do (for example, burning a small amount of data to a disc on a PC takes about a minute or so).

In addition to this, being able to write save data to a DVD would open up the console to some security issues, as someone might be able to craft a malicious save file that messes with your console or game.

Also, while DVD drives DO have storage, they only have enough for the code to read discs, with perhaps some room for some kind of update routine (e.g. if Microsoft needs to fix a bug in the DVD drive), so there's no chance your save games are stored on the DVD, or the DVD drive.

Instead they'd be stored on the hard-drive that's inside the Xbox 360.

As the other answers state, the Xbox 360 stores save data on an internal hard drive.

The potential problem is the two disks being from different regions. Many game systems consider different regions of a game to be different games, so you may not be able to access your save data with a game from a different region.

This does not take into account whether your Xbox will even be able to play the game from the different region, as Xbox 360 games are region-Locked without a modded console.