Historical societies aren't just focused on acquiring historic objects -- they're interested in the history behind the items. This is why many groups require documentation to authenticate the historic value of artifacts. Don't worry, you're not expected to cough up a receipt for a 1930s medicine cabinet. But you may be required to sign a gift deed and a statement regarding the piece's authenticity before a historical society will accept it.
Of course, documentation isn't everything. Handed-down family stories are often more valuable than any receipt or certificate of ownership. They can turn an otherwise unremarkable old object into an important teaching tool. There's a difference between a butter churn and the churn that your great-aunt Martha worked until her fingers bled to make enough butter for the homecoming feast celebrating her brothers' (your great-uncles') return from World War II. Family stories help people understand both an object's purpose and its significance in a particular time and place. Stories make the past come alive, which is the ultimate goal of any historical society.
If the heirloom is authentic and related to the society's focus, chances are the organization will accept your donation. However, items like books might be rejected if the society already has multiple copies of them. The society might also turn down items that are particularly crass or distasteful. If it was offensive back then, chances are it hasn't improved with time!
Most rejections happen because items are simply offered to the wrong society. A Nov. 20, 1863, copy of The New York Times containing Abraham Lincoln's Gettysburg Address might be of interest to the New York Historical Society or the Adams County Historical Society (which includes the town of Gettysburg), but it wouldn't be a good acquisition for the Atlanta Historical Society. It's an important historical artifact, sure, but it's not related to the mission of the Georgia-based group.
You should feel good about donating your heirloom, regardless of the organization that accepts it. Your family is an important part of that artifact's history, and your heirloom will stand as a testament to a bygone time.