[froggo]

The Sun Dance Ceremony has been performed for thousands of years, but was outlawed during the attempted cultural genocide of the Native American people.
It is a ceremony which has an intense aspect to it, as the sundancers pierce their chests and fast for 3-4 days, they tie themselves to an alter from the piercing and sway in prayer as the flesh slowly breaks free.
This is done as a sacrifice to get nearer to Visions from the Creator, to be more in tune with the wishes of the Creator
While it is common now that you see many non-indigenous people showing solidarity with Native American culture, by participating in sweat lodges, sharing circles, potlatches(which were also once banned), it has been requested by the Elders that non-indigenous people refrain from attending the Sun Dance ceremonies.

The Wi-wanyang-wa-c'i-pi (Sundance Ceremony): The only participants allowed in the centre will be Native People. The non-Native people need to understand and respect our decision. If there have been any unfinished commitments to the sundance and non-Natives have concern for this decision; they must understand that we have been guided through prayer to reach this resolution. Our purpose for the sundance is for the survival of the future generations to come, first and foremost. If the non-Natives truly understand this purpose, they will also understand this decision and know that by their departure from this Ho-c'o-ka (our sacred altar) is their sincere contribution to the survival of our future generations.

It would be common to assume that the non-Native peoples who were ignorant of the traditions that witnessed a ceremony such as this might be overwhelmed and express their concerns in a negative light. However, the concern of the non-Native people is irrelevant to the native americans when performing this ritual.
Native Americans come from a culture of Respect as being one of the primary motivators of life; their ways were not respected, and it is apparent that our white cultures tend to favour contentiousness over a sense of humility.
To witness a Sun Dance, it is essential that all be present with a humble consciousness, and if non-native people are forming opinions in their mind, instead of listening to their hearts or the earth, then that energy could negatively impact the ritual.

Here is a brief video summarizing the recent history of Sun Dances.

If you want to witness an entire Sun Dance (because lets be honest, even if you were to watch a 4 hour video of one online, you still would not feel the power) if you are not native, you will have to live with the fact that it is not yours to experience.

[Potemkin]

First we took their land for our own, and now we're trying to take their culture for our own. The tribal Elders were right to stop this. The survival of the tribe implies maintaining a difference between the tribe and outsiders. Without that difference, what is there left to preserve?

[froggo]

It's kind of like saying "you have your white man privilege, well here's our privilege."

[Potemkin]

It's kind of like saying "you have your white man privilege, well here's our privilege."

Exactly @froggo.