If the museum accepts your donation, the paperwork is simple. You sign a Deed of Gift, and the heirloom becomes property of the museum. If you've placed a high value on the heirloom, you may need to complete an IRS form to claim it for tax deduction purposes.
But you should ask some questions before signing the papers. Will you have to ask permission or pay a usage fee to reprint family photographs donated to the museum? Will family stories become museum property, prohibiting you from publishing them elsewhere? Will your stories appear in museum Web sites or other material?