Does Master Chief die at the end of Halo 3

Master Chief does not die. After completing Halo on any difficulty, if you wait for the credits to end (might want to get something to eat and drink while waiting, they take ages), and if you do not skip it afterwards you will see the final cutscene. It depicts Master Chief floating inside the Remaining half of the frigate Forward Unto Dawn. The replacement ring collapsed upon firing and it's blast radius was only big enough to wipe out the Gravemind and damage the Ark such that all flood parasite forms were destroyed by the collapse or by the sentinels, so he is still alive but is drifting in space. He enters a cryogenic state, similar to the state he left at the beginning of Combat Evolved. The ship is then caught in the gravitational pull of another planet. It is most likely a shield world similar to Onyx, or it may be a totally new planet which will be a possible feature of Halo Chronicles or another possible sequel from 343 industries. HE IS ALIVE!

EDIT: to make it even cooler, the planet he is on is his training planet, if you read the first halo book. This planet is Sigma Octunas IV. In the books, this happens, and it continues by him exiting cyro-sleep, walking on the planet and finding his training officer's corpse, Chief Mendez.
The Master Chief does NOT die. You see, when he, Cortana, and the Arbiter escape from the destruction of Installation 04B, the Forward Unto Dawn only makes it half way through the Portal of Voi. When the Arbiter went to the front of the frigate, the Chief was delayed when he almost fell off the back hanger. After he got back on his feet again, he didn't have enough time - Halo had already activated and began to shake itself to pieces, causing a massive explosion. At the ceremony to follow, only the Arbiter is there because only his half of the frigate escapes through the portal. But, the Chief's half was actually not destroyed, it just didn't make it. It's now floating through space, unknown to everyone else. He is presumed dead, but he's not. He's actually headed for an unidentified nearby planet. Whether he dies in the atmospheric entry is still not known, but Master Chief Petty Officer John SPARTAN-117 does have a lot of luck on his side. Cortana, however, is doomed to 'die'. Artificial Intelligences, AIs, have a period of time in which they function and show personality, like they're real people. The average AI 'life' expectancy is about seven years. Cortana's already past that mark, and almost went 'rampant' as it's called when AIs stop working when she was aboard the Flood-occupied High Charity. She was almost rampant, but she wound up being discovered again by the Master Chief. But, if he does die on this unknown planet, it won't be in Halo 3. So, the answer to this question is: no. He doesn't die.