Indeed, the question given in the title does not make sense without the prior definition of what is a "board game", and the latter appears to be quite arbitrary and unsatisfying whatever restriction(s) would be chosen. To illustrate what I mean, consider that even the most "basic" requirements given in earlier answers are unsatisfying: 
(1) The finite number of possible constellations: as mentioned somewhere, one could well imagine pieces which grow in (whatever) "strength" without limit. (From the piling up of pieces to denote kings in checkers, its only one step further to imagine a kind of promotion that allows a player to attach an arbitrarily large number to any of his pieces). 
(2) A move = piece taken from one location to another, possibly removing another enemy piece: obviously, an arbitrary number of enemy pieces might be removed (as in Go).
 - Also, to include moves similar to castling in chess, the definition must potentially allow several pieces from different locations to move at once into possibly any other location.
 - To include promotions, pieces must be able to be changed into any
   other of the (as said earlier, possibly infinitely many) pieces.
(3) Also, many of the most popular board games involve throwing dice (which was the case for chess for a long time, not so long ago), asking and answering questions to the other player(s), ....
So, in short, it appears that
(a) a "state" of the "board" most comprise more than a finite number  of locations, possibly infinite number of possible pieces, and other accessories like dice, clocks, infinite collections of questions and answers ...
(b) a transition from one such state to virtually any other state of the board, must be compatible with a general definition. 
We see that the physical board and the "currently active pieces placed on that board" are only an   infinitesimal part of a possible "state". So, whatever restriction could prevent *any* game of being a *board game*, according to any  satisfyingly general definition of the latter (i.e., compatible with the most popular board games) ?
We should admit that on a closer look, the "board" itself is, in spite of the first appearance, an (almost(?)) negligible part of most board games. And any non board game could probably be a board game with a possibly trivial (empty or singleton) set of "locations". 
So I wonder whether the notion of "board game" can make a strict logical sense (as opposed to the obvious intuitive sense), unless of course we don't require it to be compatible with real-world chess, Go, Chinese checkers and common popular board games using dice.

[Edit: Also, multi-player games must be comprised in a satisfying definition, as to include the very classical board games of Halma and Chinese checkers.]