Playing SWAT 4, I needed to know what to do before going in and failing the old fashion way, so I went into the training mission and finishing up the shooting range portion entering into the metal silhouette portion where they pop 'em up to knock 'em down, I noticed the instructor in the catwalks above and decided to shoot him. Lo and behold, he ragdolls and I failed the mission. Objective: Do not kill Lieutenant Bonds FAILED.

What other tutorials allow you to just casually fail them? I found it hilarious that not only does SWAT 4 allow you to just shoot the guy, but he ragdolls upon death. A different kind of game would either not allow you to shoot them at all, or simply kick you to the game over screen and then they are still standing there.

These two might not be called tutorials in the usual sense, but not only is it possible to fail them, but it's reasonably likely that you'll fail them because of bad RNG.

Final Fantasy Mystic Quest:
* At the very beginning of the game, during the intro cutscene, you will have to fight a Behemoth. Your options here are very limited, as you have only one party member, and no items or spells, so you can only attack, defend, or run away (which will never work since this is a scripted battle). Defend isn't going to help, so the only thing you can do is attack. If you miss too many times (bad RNG), or if the enemy scores a critical (bad RNG), then you are likely to lose the battle. Fortunately, after you die in FFMQ, you are given the option to restart the battle from the beginning; just say "No" when asked if you want to give up. (This ability to retry after a party wipe applies to every single battle in the game.)

Shin Megami Tensei: Nocturne (hard difficulty):
* It takes a while to get to the first battle (and there's fortunately a save point shortly before it), but once you do, it's again RNG dependent whether you win if you're playing on hard. There is a good chance that the enemy will score a critical, taking off over 2/3rds of your health, and due to the press turn system the game uses, the enemy gets a free turn when this happens, allowing it to take of the remaining 1/3.

In Metal Saga, it's possible to get an ending in the intro before you gain control of your character; this ending tells you about all the awesome things you didn't do because you decided to become a mechanic rather than a hunter.

It's also possible to die in Celeste's tutorial; there's a falling block that can crush you. Then again, you have some later tutorials, including one in Chapter 9 that teaches the player some very advanced tech, and you have to pass it to continue. Chapter 8 teaches the player the level's gimmick by presenting the player with a screen that would normally be easy, but when you try it, you discover that you don't get your dash back when you expect, and end up dying as a result.

(It probably doesn't count, but Final Fantasy 2 opens with a battle that you can't win; losing is required to progress. In fact, on some versions, winning can result in the game rebooting or crashing, as the developers didn't bother to account for this situation which is impossible without hacking.)

Edit: Why the low rating? What did I do wrong here?

I feel like you have to try, but I imagine that it'd be possible to flunk the Tutorial Well in Glover.

Lost Vikings 2 is aware that you can flunk the tutorial, and it even rewards you for doing so, since you'd have to go out of your way for the setup.

Police Quest is an annoying snooty bastard, in that the entire game may as well have been a copy protection course. If you fail to do anything by the included book, you'll likely die or fail.

Reminds me of Final Fantasy Tactics, where it is possible to lose the first battle. However, this is extremely unlikely, as it turns out to be rather difficult even if you try to do this on purpose. Thing is, you have allies who are of a higher level than you, and you don't control them; furthermore, it looks like the enemy AI will attack your allies and not you, unless you're the only one alive.

In any case, this is something that's theoretically possible, but rather unlikely.

Edit: Again, why the low rating?

The "map/artillery/ airstrikes" section of the "Vietcong" tutorial.

Apparently I wasn't really paying attention, when the game explained how to call in artillery strikes, since I managed to misinterpret the map completely, and fired on my own position.

In utter disregard to what had just happened (obviously enemy artillery wiping out my position, before I could wipe out theirs), I targeted the same square again in my next attempt to pass the tutorial - simply hoping for a different result this time (and we all know, what that means).

To shorten the story: I actually managed to repeat this several times(!), until I finally noticed that you can (and have to) scroll the map to see the actual enemy's position.
To add insult to injury, the manual says this (quote):

"Select where you want the attack to land by pointing your cursor on your desired spot and press the fire key.
Boom.
Hope you didn't kill yourself."

:/

I think the most famous example could be beloved, unskippable training mission in Driver.

https://www.youtube.com/watch?v=kmRPTt0588g

Possibly the hardest tutorial of any game ever.

So satisfying when you get through - although the entire game excepting the last mission is significantly easier than the tutorial!

Maybe not the "tutorial" per say but I enjoyed killing Captain Keyes in Halo upon first awakening. Killing him alerts the ship that Master Chief has gone rogue and you get assaulted by soldiers with incredible aim to take you down while all you have is a pistol. I guess these guys died when crash landing on the ring?

Geez, no wonder they went on to make the first Stuntman.

Makes me think back to True Crime New York's tutorial segment, which has vehicle training, but before that you're tasked with choke holding some training volunteers, which you're straight up allowed to snap their necks. This won't fail you, but the instructor will berate you and give you some bad cop points. (I actually kind of want to revisit that game since apparently the PC version was somewhat cleaned up compared to the buggy and incompletable Xbox version)