If you buy the new Note 8, you need to buy the latest Gear VR headset, because it doesn't fit older Gear VR headsets. Some people might modify it to make it fit, but it personally wouldn't do that. According to an article that I've read on the Verge, Samsung announced that it's bringing a new Gear VR headset model that fits the large 6.3-inch screen of the Note 8.
It's kind of weird of Samsung not thinking about it ahead of time. Why release a new headset if you've just recently released a new Gear VR headset. The good news is that if you buy the new Gear VR headset, it will have backwards compatibility with all the previously supported Galaxy devices, which includes the S8, S8+, S7, S7 Edge, S6, S6 Edge, S6+ and the Noet 5. I personally think that Samsung should have offered the new Gear VR headset for free for those who buy the new Note 8, but I haven't found anything about this kind of offering. If you remember, Samsung offered a free Gear VR headset for those who purchase the Galaxy S7 and if I am not wrong, the S8 as well.
I don't think it's a big deal, and people who will gladly pay for the Note8, probably won't have a problem shelling a bit more and get the new Gear VR headset in order to consume VR content using their phone. That being said, the previous Gear VR headset with the controller costs $93.99 (as of August 24th, 2017 via Samsung's website). It's not a cheap price to pay and now you can see why getting the Gear VR was such a good offer back then.
I don't know about Daydream support though, but I guess the Note8 will support it as well. Even then, I'm pretty sure that people who buy a premium Galaxy phone will definitely want to try out Samsung's own mobile VR before trying out other platforms. By the way, the Gear VR 2016 version without the controller costs $40 now, but as I mentioned, it doesn't support all phones, it's only compatible with the following Samsung Galaxy smartphones: Galaxy S7, Galaxy S7 edge, Galaxy Note 5, Galaxy S6, Galaxy S6 edge and Galaxy S6 edge+.
In this video by phonearena you can check out the new Samsung Gear VR that was designed to also work with the Galaxy Note 8. Nice hands on videos so check it out.