The studio also notes that due to the age of these games and their services, 'access to online functionality even prior to the official closure date is not guaranteed.' Servers for FH1 and FH2 have been all over the place in recent years, so being able to get online and pop achievements (that aren't already discontinued) in both games will probably be a bit of a challenge.
'While the single-player experience in both games will remain playable, any online functionality such as matchmaking, leaderboards, and the browsing and downloading of online User Generated Content (UGC), including liveries, vinyls, tuning setups, and photos will not be available from August 22, 2023,' Turn 10 says. The server closure impacts both the Xbox 360 and the Xbox One versions of Forza Horizon 2 as well as both the Xbox 360 and Xbox One versions of the standalone game, Forza Horizon 2 Presents Fast & Furious, though there are no strictly online achievements you need to worry about here.