The instant Death by Cryo Tube trick is available only for Halo: Combat Evolved.
Just after the first cutscene in the Pillar of Autumn, you begin de-thawing from a cryo tube. Right after you jump out of the "hushed casket", jump on the tube so that you are under the cryo tube's cover.
When the cover closes, you will be pushed "through the ground" and "fall to your death".
Sometimes the cover will not push you through, but instead will close and trap you in the cryo tube. If you turn away from the cover you can see a headless Master Chief laying in the tube. You can do this by jumping on the tube and crouching before it closes. Look down and you would see what appears to be clones of the Master Chief on the ground for 3 seconds.