The pickle dish is significant for several reasons. The first is that it is a fancy dish that was very special to Zeena. She treasured it, because it was a wedding gift, and she only brought it out on special occasions. That is why she kept it high up on the shelf--so that it couldn't get ruined or broken.
The second reason that it is significant is because it is a symbol of Mattie and Ethan's rather forbidden and socially unacceptable romance. Getting the dish down for a special dinner for Ethan was a bold statement by Mattie--it revealed that she truly cared about him. It was her stepping into Zeena's role as his wife and provider, wanting to take care of him and provide a nice dinner table for him. It was not her place to do so; it was not proper. And in doing so, it became broken.
The dish being broken would reveal what Mattie had done, and that might clue Zeena in on the relationship between her and Ethan. Why would Mattie get out the dish just for Ethan's dinner? It is a very revealing clue, and one that leads to Zeena wanting to get Mattie out of the house. The broken dish also could symbolize Mattie and Ethan's broken relationship at the end, their broken bodies, and even the broken marriage vows between Zeena and Ethan.
So, the dish is significant on many levels, both symbolic and literal. I hope that helped a bit; good luck!