Rooting your brand, new Galaxy Note 3 will most likely void your warranty.
But there’s also ways to “unvoid” warranty and in most cases, you will be able to “unroot” your Galaxy Note 3 completely back to stock without carriers knowing you’ve rooted your phone.
These methods are not readily available yet but as open-source developers start to work on the Note 3, there will be methods to unvoid your warranty easily.
Of course, sometimes carriers (such as U.S. carriers) do not even check to see if you’ve rooted your Note 3 as their primary concern is with hardware. They will most likely check for water damage and so long as you have stock Samsung firmware, you will be able to use your warranty fine. (E.g. I’ve personally sent a Sprint phone with custom ROM but they got it replaced with no questions asked. So it may depend on your local store but most places are not too strict about rooting.)