In a small saucepan over very low heat, melt the butter. After the butter has melted, skim off all the foam. Add the tangerine zest, juice, and salt. Pour about ½ cup of the butter mixture into a small cup; reserve the rest for the dipping sauce.
If using bamboo skewers, soak in water for at least 30 minutes.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Using kitchen shears, split each tail in half lengthwise. (If desired, to prevent the lobster meat from curling during cooking, thread a skewer lengthwise through the center of the meat of each piece.) Lightly brush the exposed meat with the butter in the cup. Brush the cooking grates clean. Grill the lobster tails, meat side down, over direct medium heat, with the lid closed as much as possible, until lightly marked, 2 to 3 minutes (watch for flare-ups). Turn the tails over and grill until the meat is white and firm but not dry, 5 to 8 minutes longer, brushing occasionally with some of the butter in the cup. Remove from the grill and serve warm with the reserved butter on the side.