No Princess Zelda is not dead.
Princess Zelda was well and alive in the latest installment in the Zelda series The Legend Of Zelda: Breath Of The Wild. The reason why there are a lot of rumors in the gaming community about whether Princess Zelda is alive is that throughout the whole series the developers have opted to have Princess Zelda go through a lot of flashbacks which may mean that she is not alive theoretically although you have to be alive to have flashbacks in the first place.
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The Princess Of Zelda is often confused with Link as he is the poster boy protagonist of the game called Zelda as we all know. There are various incarnations of Princess Zelda, in some games, she is more playful and flirty with Link while in others she is much more reserved and serious. Her various versions are also portrayed to be great musicians with at least three of the incarnations of Princess Zelda being great harp players while others being known for playing other instruments such as an ocarina or a pan flute.