I don't know if it's possible (if it is, then it's hard and you will need multiples arduino)
You could glue a magnet under each chess pieces. To know if a case is vacant or used, you could use a few Hall effect sensor positioned on top of your Led array.
I don't think it's important to filter the surrounding magnetic field.
Then, maybe it's possible with trigonometry to actually detect where the field is stronger (with the multiples Hall effect sensor) and get a somewhat clear position, just use the same magnet model/type for each pieces.
With RFID you could assign the correct chesspiece an ID so the arduino know what are the pieces. If you can write in the sketch some basic rules of Chess (aka, game grid, start position and chess pieces) Maybe it could actually follow the mouvements of each pieces from the start to the end of the game.
This is just an idea, all theory...keep in mind that I'm a complete n00b and I don't own any Arduino.
or Just use push buttons, more simple.