Start a game, bring up the console with ~
Enter: con_restricted 0 to unlock the console commands
Enter: cl_hud 0 to disable the hud
if you're playing the demo, enter: r_displayinfo 0 to disable the text that says "DEMO" in the corner of the screen.
This disables the hud AND the crosshair. I do not know how to only disable the hud. Luckily this game has an iron sights option.
To make the setting permanent, exit the game and open your Crysis\Game\Config\singleplayer.cfg:
enter the following commands:
con_restricted = "0"
cl_hud = "0"
r_displayinfo = "0" (only necessary in the demo)
To disable the intro movies in Crysis, change the name of your Crysis\Game\Localized\Video directory. I just renamed it Video.bak.
If you want to keep specific videos, but not others, just open the directory and rename only the files you want to disable (I find just adding an extension like '.bak' works well, so that I can still tell what the original extension is.