Would a magnet interfere with NFC devices in this scenario?
No.
NFC relies on alternating magnetic (and electrical fields) changing 13560000 times per second.
A magnet represents a steady field, or at most one changing at the comparatively low rate at which you could physically move something.
There will be no interference in terms of the actual signals.  And for reasonable magnetic field strengths, eg, things you could actually purchase and manually move, there will be no overloading of the components to the point where they misoperate.
It's not entirely certain however, that you'll have sufficient isolation between your various chessboard squares for NFC to uniquely detect position.  It may work with care, but arrays of NFC antennas aren't quite the usual case, so there may be some engineering there, as well.