Does the Xbox One re download games if they were previously installed from physical discs?

If you have already installed the game from disc, then the xbox will not need to download the game again if you purchase it from the xbox store.

So just to be clear, if I do purchase it on the store having already installed it, it
will not download anything and I will no longer require a disc.

Although to play the game that was installed from the disc you have to re insert the game every time you play the game. If you are wondering if you game save will stay with the game you played from the disc when you switch to the downloaded copy, it will.
example, I had FH1 on my Xbox 360 via disc... lost disc... purchased the game when it was free. game save stayed with me and played even when I was on the digital download copy.-- same applies with Xbox one.

So if I were you go to games and apps> go to FH3 box> hit menu on controller> hit manage> Delete the Game but not the game save from the cloud. (do this if you don't care about the internet connection-- if you do care attempt to purchase the game and play
the title as Stuart says)

After you purchase and download the game you will be able to play Forza Horizon 3 with out having the data loss from disc game

Hope that helps

If you have already installed the game from disc, then the xbox will not need to download the game again if you purchase it from the xbox store.

To explain a bit why that won't work:

The Xbox One doesn't understand the "language" in which the Xbox 360 games are stored on disc or USB. The disc for physical copies is just a "proof of purchase" for the Xbox One, and the actual code for running the game on the Xbox One is downloaded from
the Xbox Marketplace in a "translated" language the Xbox One can understand.

If you installed a game from disc on an USB drive, it's missing the "proof of purchase" license the disc would provide.

If it were a digital download game, it would show up under Ready to Install in your account's Games and Apps, and you could simply download it for free, as the "proof of purchase" for a digital download game is stored in your account.

However without a disc, there is no proof of purchase, and you can't install the game on the Xbox One.

Are you playing the game from the disc, installed to the HDD from a disc, or downloaded from the Marketplace? Can you try deleting the game (including any Title Update files), clear the system cache, restart the console, and re-download or re-install the
game. Does this help?