I know nothing of the board game, but Googled "Billie Burke board game" and came up with this photo:
http://upload.wikimedia.org/wikipedia/commons/f/f1/Billie_Burke_Game_B oard.jpg
Hope you can read the printing better than I, it has some information, although not a lot.
I just found this blurb from the Wiki file that holds this jpeg:
Billie Burke is depicted on this 1905 Game board titled Amerst which is included in a Boardgame/Toy published by the Geo B Doan & Co. titled The Twentieth Century Fun Box. No acknowledgment is made to this game title in any popular collectors value guides or internet search. The game manufacturer is recognized for other work. I welcome interest to this rare item and will forward a full board photo to anyone interested.

I was wondering if you could give me any information on the game, like, the title of the game, who made it, how it's played, or where you found it.
My Father and Uncles insist that we are related to Billie, but I have no way of proving that at this point. Their parents died when they were very young, and the only family information I have, (other than the potential Billie Burke connection) are their parents names. I know their Mother's name was Virginia Lee Burke, but I have not been able to locate any birth or death records for her or their Father.
Regardless, I have always loved Billie, I think she is a classic beauty and there is just something remarkably special about her. I've seen the Wizard of Oz more times than I would like to admit!
I have collected a few memorabilia items associated with Billie, (mostly small, inexpensive items like lunchboxes, etc.) and I would love to see the board game you are referring to. I have never heard of anything like that, so I wouldn't know where to even begin looking for it!
Any information you could give me about the game would be greatly appreciated!
Thank you!

What kind of board game is it?
Team
You've Got A Very Important Date