Iphone
If I remove my old iPhones account from my new iPhone that someone else is currently using does that remove their data from their phone
When changing to the new ID, there may be some data loss, but you can minimize it. If syncing photos with iCloud, ensure Optimization is turned off in Settings/Your Name/iCloud/Photos and/or in Photos/Preferences/iCloud. Then make sure full resolution photos are on the device/computer. When signing out of iCloud, if asked whether you want to retain data, choose yes. Messages conversations will be lost if they are addressed to the ID. See the article below for other places that the old ID should be removed. See Winston Churchill's response in the below discussion.
How to switch Apple IDs on your iPhone or iPad.
How to transfer all files from one iCloud account to another.
What to do after you change your Apple ID or password - Apple Support.