Do the 2001 Dodge Neon door handles open the door if it is locked?

yes

Not from the outside!!!

If you have a sedan and the child safety latch is on, you won't be able to open the back door from the inside even if is unlocked. Otherwise, all doors can be opened, by pulling the handles from the inside, when they are locked. Of course, you won't be able to open them from the outside while locked, unless you have the key, or the remote!

No matter the position of the child safety latch, it is not possible to open the back doors from inside when it is locked. That mechanical auto unlock feature is only for front doors.