I foresee downloadable board games, Sites springing up, like open-source-boad-games.com, where people post an ad for their board game, along with the rules/how to play, and include a download link for the 3D Objects as needed by various 3D printer applications.
I can also imagine Pausing a game (or a game in replay) and selecting the key parts, like two characters in a clash. Then instead of saving the screen shot, you save the scene in a 3D printer compatible format. And part of the game is to actually help you make really awesome 3D action models/scenes.
I don't foresee this as large scale, but heavy enough for major board game companies to feel worried, though not at a significant loss. Marketing and shelf space is a huge part of how games sell successfully.