My son scratched his Forza Horizon disc and now it keeps freezing on him. If we buy another disc, will all his data and progress be lost?
All optical discs for consoles of the 6th, 7th (includes Xbox 360) and 8th generation are read only media as far as I know. Nothing can be stored on these discs after they have been created in the factory. The stock optical drives in these consoles are also missing the capability to record or "burn" on such media even if you use writeable discs.
So a "savegame" that stores the progress has to be stored somewhere else. Earlier generations used special memory cards for this purpose. The first Xbox introduced a hard drive in the default configuration, with the 360 memory cards (earlier models) and hard drives were optionally available (to reduce the price at that time). On later models of the 360 a portion of internal flash memory was available for additional game content and savegames. USB drives have almost replaced memory cards and Xbox Live Gold offers to save progress to online storage, which is the default on Xbox One.
Apart from different locations where progress can be stored on current consoles you should make sure that the replacement disc is compatible and not a different edition of the game or specific to another region. This is a rather rare case, but for example I have DOA5 and DOA5U which are detected as two completely different games where achievements are exactly the same but both games have separate entries. The newer edition knows about the older and can import progress, but this functionality is completely up to the game manufacturer to implement. (To add to the confusion DOA5U was patched to DOA5LR. :D) More related to Forza, Forza Horizon 2 is available on Xbox 360 and Xbox One, but both are completely different games and even though you can save your progress online in both games, they're not compatible.
I don't see any issues with Forza Horizon in your case, but as suggested, trying to resurface the disc is the first option I would try.