Due to the way that PlayStation Network was originally built, and the way that Sony implemented PSN ID changes across the last few years, there are potential issues with certain PlayStation 4 games if you choose to change your PSN ID.
Sony slowly rolled out the ability to change PSN IDs — which was first announced last year — so that it could test for potential issues. What kind of issues?
Here's what Sony says:
-Your previous Online ID may remain visible to you and other players in some areas.
-You may lose progress within games, including game saved data, leaderboard data, and progress towards Trophies.
-Parts of your game and applications may not function properly both online and offline.
-You may lose access to content (including paid-for content) that you may have acquired for your games including content like add-ons and virtual currency.
Not insignificant stuff! If you're worried at all, there's a whole list of games with more specific information right here.