Place the beef tongue into a tight-fitting pot and cover with water by 1 inch. Add the salt and bring to a boil. Reduce heat to medium-low, cover, and simmer until the meat is tender, about 2 hours. Strain and reserve the cooking liquid. Remove and discard the skin from the tongue. Shred the meat using two forks; set aside.
Heat the vegetable oil in a large pot over medium-high heat. Stir in the onion, leek, celery, and carrots. Cook and stir until the vegetables soften, about 5 minutes. Stir in the tomato paste until no lumps remain and cook for 1 minute more. Add the shredded meat to the pot along with the reserved broth and taco seasoning. Bring to a boil; reduce heat to medium-low and simmer until the vegetables are tender, 20 to 30 minutes. Season to taste with salt and pepper before serving.