Is it possible to force the DHCP server to assign a different IP address each time the address is renewed?

By definition a renewal attempts to renew the existing lease of the same IP address.

If you want to force a new IP address from the client side you can change the MAC address of the network adapter. The DHCP server will see it as a new client and allocate a new address.

You may want to look at the macchanger package http://manpages.ubuntu.com/manpages/bionic/man1/macchanger.1.html as it has some nice to have features in this regard.

macchanger is a GNU/Linux utility for viewing/manipulating the MAC address for network interfaces."

If you want to force a new IP address from the server side you can change the configuration such that the current IP address is no longer available. This can be done by creating a reservation allocated to another (possibly fictitious) machine/mac address, modifying the scope so that the old address is no longer contained within the scope, disabling the scope or server entirely so that the server does not respond, or other methods that prevent the server from issuing the renewal.