In an email to museum staff, Mr Rankin said he wanted the items removed from display and returned by this Friday.
"You ... have behaved in a culturally reprehensible manner over this loan, and it is clear that our taonga are no longer in a culturally safe space," Mr Rankin wrote.
"Considering that Heke was the first rangatira to sign the Treaty, this has been regarded as extremely offensive by our kaumatua and kuia."
He said he would refer the matter to the police if necessary.
Waitangi National Trust chief executive Greg McManus said it was difficult to comment at short notice and without speaking with staff.
He was confident that loan documentation would have been completed for all loaned items.
"If there has been an issue with Mr Rankin receiving these I can only apologise and would like to discuss it with him next week."
The Museum of Waitangi features taonga, artefacts and around 500 images from private collections and museums around the world. Some of the items are nearly 200 years old.
There is a permanent exhibition at the museum called Ko Waitangi Tenei: This is Waitangi, which explores the stories behind the place and its people.
There is a copy of the first bilingual Treaty -- a document made by missionaries in 1844 to aid understanding between British and Maori. But it later led to conflict.
Mr McManus said the dawn blessing of the museum was conducted by senior kaumatua and church ministers from throughout Ngapuhi, and -- because it was open to all -- no invitations were sent.
He said the ceremony was widely publicised among Ngapuhi and some 250 people attended.