This seems to bring up a lot of debate in individual game topics whenever someone wants to do a run with 'NG+' mechanics - using pre-existing save data to start a new game, but some data erroneously mixed in from the save, allowing for a faster run that is still 'new game'. I think a general ruling should be made for how such runs should be treated - are they technically any%? A new category? Should not be accepted?
To facilitate this I'm going to post as many examples of 'NG+' mechanics as I'm aware of in speedruns. Feel free to post ones you know of too.
Donkey Kong 64 - The 'Funky Weapons Glitch' requires a cart where you've EVER beaten the game on - even if you erase all save files, it still works. It lets you start a new game with the abilities of a multiplayer match already unlocked - weapons and sniper scope, notably. There is also a way to partially glitch completion from a save file onto a new game run, which is only allowed in 106% glitched/max% glitched categories and not in any%.
https://kb.speeddemosarchive.com/Donkey_Kong_64#Funky_Weapons_Glitch
Ocarina of Time 3D: The category called 'NG+' allows you to start a new game with any item glitched onto B - popular choices are farore's wind for any% and megaton hammer for MST. This is done by using the bottle adventure glitch on a save file to get the item you want on B on B first.
http://zeldaspeedruns.com/oot3d/ng/creating-a-ng-file
Crash Bandicoot 2: A glitch discovered today allows you to start a new game in warp zone 4, the category has been called NG+ (NG+ any% or NG+ 100%).
Castlevania: Symphony of the Night: It is possible to transfer the level of the sword 'Muramasa' from a save file to a new game run. I don't know of the specifics, just that ForgoneMoose & arukado plan on using it for the new any% TAS and it has apparently been OK'd for use.
Final Fantasy: Mystic Quest: The 'resistance bug' lets you start a new game where character 2 keeps some traits of the equipment of character 2 from a previous save file. It is used in RTA categories.
Diddy Kong Racing: It is possible to transfer the fact that Future Fun Land (the last area of the game) is open from a separate save file then, from a new game, use OoB glitches to immediately go there and beat the final boss. A TAS of this category was rejected:
http://tasvideos.org/forum/viewtopic.php?t=13743 http://tasvideos.org/forum/viewtopic.php?p=338326#338326
Densetsu no Stafy: It is possible to transfer the state of a save file over to a new game, putting you at the final boss on a new game and allowing you to immediately win. A TAS of this category was rejected:
http://tasvideos.org/3832S.html
The Legend of Zelda: Skyward Sword: The glitch 'Back in Time' makes it so the intro screen is playable. In this state you can also access the menus to load and save, while saving using in-game save points (because you have made the intro screen playable), allowing you to corrupt two save files together. Using this and a pre-made save file you could easily go from the first BiT to the end of the game, but all speedruns disallow using a save file made before the run has started in BiT.
Super Metroid: The post by ais523 explains this one better than I could:
http://tasvideos.org/forum/viewtopic.php?p=338326#338326