If someone logs into their Apple ID on my phone and downloads an app would anything happen to my phone?, If someone logs into their Apple ID on my phone and downloads an app would anything happen to my phone?

This will not damage your phone but it might cause problems later. Apps downloaded or purchased are forever associated with the Apple ID used to download them. Updates to the app will require the Apple ID and password used to download the app originally.

I suggest that you delete apps downloaded with an Apple ID that was not yours. This will avoid the problem of requiring that ID when the app needs updating.

Then make sure that you are still not signed in to the other Apple ID. Use the Settings app for this.

Settings > iTunes and App Store > what Apple ID do you see there?

If the ID is yours that is fine.

If the ID is the wrong one tap in the ID and then tap on Sign Out. Finally sign in with your correct Apple ID and password.