Single player content will continue to work.
Screenshot of Forza Horizon 1.

Forza series developer Turn 10 Studios announce that the online services for both Forza Horizon 1 and Forza Horizon 2 will be going offline on August 22, 2023. They say that this is due to both of these titles now having "a very small player population." This move will allow them to "focus on (their) most current games."

The good news here is that the single-player content for both of these titles will remain playable without issue. However, content like online multiplayer, matchmaking, leaderboards, and browsing/downloading user generated content (liveries, vinyls, tuning setups, and photos) will no longer be available after August 22, 2023.

We know our community has fond memories of exploring Colorado and Southern Europe in Forza Horizon 1 and 2, from meeting new friends on road trips to sharing liveries, photos and tunes, and we thank you all for your incredible passion, support and dedication over the last decade.
Those still playing Forza Horizon 3, 4, or 5 don't have anything to worry about with this closure. It's also important to note that even though the online services are being shut down, this will not prevent you from redownloading Forza Horizon 1 and 2 for single-player gaming in the future.