No, there's a good chance you do not need a static IP address for the purpose of port forwarding for hosting games.
The way in which port forwarding works is, in essence, you have a series of application layer ports accessible from computers on your lan by default. However, as these ports are only local, they will need to be "forwarded" and also possibly placed in a demilitarised zone depending on your router for the ports to be visible to the outside world.
Your ip moving around could be an issue, however there are some very simple workarounds for this. Take dynamic DNS for instance.
There are free dynamic DNS services available online (no-ip etc) which will update a domain name on a basis as frequently as you choose with the latest external ip of your router. From this, people will be able to enter your domain name into the game (example.ddns.net) and join to your most up to date ip. If the game does not support Ddns then you may also be able to use the service, but your friends will have to do nslookup example.ddns.net to get the ip and then enter the ip in game.