So, say I knew the wifi password to someone's wifi. If type the password in and use the wifi then press "forget network" it removes the password from my phone. Does this then remove the device and the devices IP address from the router so the admin can't see it??
No. There is no such protocol to "remove the device" and its IP address from the router. "Forget network" only takes action on the local device by removing whatever data is stored about the network.
You could potentially request for the DHCP record to be released while connected to the network using command-line utilities; it might remove the IP address assignment on the router, but that depends and it is likely that it would still leave traces.