When I think of knotty pine - two visions come to mind. Light natural knotty pine or Alder and then shellacked dark knotty pine. They are two completely different looks. I am not sure which you have, so I will give you an answer for both.
Darker Knotty Pine
If you have the darker knotty pine then I would go with a classic black counter top and tumbled tile for the backsplash in a natural stone color. Since this type of knotty pine has a classic woodsy cabin-like feel to it bringing in natural elements will enhance it. A stone tile backsplash would complement the pine and black counter top as it has a mix of natural colors – not just one flat color. Look for one that has bits of black, grey, and red or orange tones in it. If you want to paint the backsplash - look to colors that either match the pine or are a complete contrast. Field Poppy would work well. It is a vibrant modern paint color that will update the room.I am not sure of your budget, but concrete countertops would also look fabulous. They are a little harder to come by than Formica®, Corian®, or natural stone surfaces, but would be the perfect complement to the knotty pine.
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Lighter Knotty Pines
If you have lighter knotty pine on your walls and cabinets– more like Alder, your options increase as this could be considered a neutral. I think of Sarah’s house from the HGTV show Sarah’s Summer House where you could use lots of white, greens, and blues that will evoke a soft, warm, and cheery lived-in appeal. You could add bead board painted in Livin' Large, Aloe Vera, or Victoria Blue for the backsplash and have a simple white counter top.Find Your Favorite Paint Colors