Can I configure a fixed IP address on my iPhone?
It's in the network settings:


Wi-Fi 
join your network if needed
tap the blue right arrow
Select Static instead of DHCP/BootP and enter in the fixed IP you prefer


Visit http://help.apple.com/iphone/ if you want pictures to guide you to the network settings part of the settings app.

This only works on Wi-Fi since your carrier issues you an IP address. You could also use a VPN profile to tunnel all traffic to a fixed IP if you had a router / server set up to provide that sort of IP / DHCP to iOS devices.