The regulations are in 50 CFR Part 17. This is about 300 pages long, and there is no single answer – you have to investigate in a case by case manner. African elephant ivory, as an example, is covered in §17.40(e). Subparagraph (3) says that
Except for antiques and certain manufactured or handcrafted items
containing de minimis quantities of ivory, sale or offer for sale of
ivory in interstate or foreign commerce and delivery, receipt,
carrying, transport, or shipment of ivory in interstate or foreign
commerce in the course of a commercial activity is prohibited.
So it will ultimately depend on the nature of the item. Assume that the item is an "antique", which is defined (regardless of species) in 16 USC 1539(h). The requirement is that the item
(A) is not less than 100 years of age; (B) is composed in whole or in
part of any endangered species or threatened species listed under
section 1533 of this title; (C) has not been repaired or modified with
any part of any such species on or after December 28, 1973; and (D) is
entered at a port designated under paragraph (3).
Paragraph (3) says that
The Secretary of the Treasury, after consultation with the Secretary
of the Interior, shall designate one port within each customs region
at which articles described in paragraph (1)(A), (B), and (C) must be
entered into the customs territory of the United States.
So it could be anywhere, subject to change.
As for documentation, this document, which is the final rules for a change in rules pertaining to African elephant ivory, has a section Comments on documentation requirements. Their suggestion is
Provenance may be determined through a detailed history of the item,
including but not limited to, family photos, ethnographic fieldwork,
art history publications, or other information that authenticates the
article and assigns the work to a known period of time or, where
possible, to a known artist or craftsman. A qualified appraisal or
another method, including using information in catalogs, price lists,
and other similar materials that document the age by establishing the
origin of the item, can also be used.
This could address the age requirement; the port of entry requirement is mildly problematic since the relevant law was not passed 100+ years ago so there were no specified ports of entry. They do not address how one is to prove that the item was not repaired after December 28, 1973.
The regulations themselves does not specify the documentation requirements, so if you run into trouble with the Feds for shipping your piano, you can point to the final rule.