In the Arcade GameRampage, your character is a human who's been turned into a giant mutant animal. If your Life Meter empties, you turn back into a (naked) human, and your character hastily covers their naughty bits and scurries off screen.
The "Naked Raiden" sequence in Metal Gear Solid 2: Sons of Liberty. If you have Raiden stand against the wall in just the right place in the examination room where he first wakes up after losing his clothes, you'll see him without his hands over his genitals...only to find that HE HAS NONE. He's like a Ken doll! No wonder he's embarrassed.
In Elite Beat Agents, failing the first section of one song results in Mr. Virus, the rival, ripping off the clothes of Cap White, the person you're trying to help. She promptly covers her breasts with her hands (by the way, this is essentially the closest thing to nudity the game allows, but it's still shocking coming from Nintendo, a company infamous for their strict rules of censorship in the early 90's).
When standing still while playing the streaking minigame in Saints Row 2, the player character will stand with their knees together, shivering and covering his/her privates with their hands.
In Dead Rising 2, you can play Strip Poker, and the women will keep playing until they're down to their panties. Once they're topless, they cover themselves with their hands.
In The Adventures of Lomax, when you attack a werewolf, it transforms into a small naked lemming who covers himself with his hands.
In Jazzpunk, the female attendants at the Kai Tek Resort are topless with leis acting in a manner similar to Godiva Hair.
Several times during Leisure Suit Larry: Love For Sail, Larry loses all of his clothes and has to sneak back to his room naked, covering himself with an object related to the last scene.
When playing as Donald Duck in Disney's Magical Quest 3, losing a life while wearing Donald's version of the armor costume (which has Donald wear nothing but a helmet and a barrel) causes Donald to lose the barrel and cover himself in embarrassment.