I keep them in for the heat. My juice machine obliterates them so I don't have seeds in the juice. If you're using a blender, maybe straining should remove them while maybe allowing some of the heat from the seeds to remain in the juice? I am not sure.
I am not too sure how V8 Spicy Hot gets its heat, I could not find anything in the ingredients list. Jalapenos are the closest I have found taste wise (I tried chilli powder, green and red chillis and small amounts of habenero).
I should have added the beet is the most sensitive of all ingredients, it can dominate the flavor if you use too much. A spicy beet drink is no good