Heat olive oil in a large skillet over medium heat. Add beef, shawarma seasoning, allspice, and 1/2 teaspoon salt; cook, turning occasionally, until beef is browned and tender, 3 to 5 minutes.

Whisk tahini, yogurt, lemon juice, and 1/2 teaspoon salt together in a small bowl until creamy.

Spread 1 to 2 tablespoons tahini sauce in the center of each tortilla. Divide beef slices, tomato slices, and onion slices evenly among tortillas. Roll up tortillas.

Preheat a panini press according to manufacturer's instructions. Press rolled tortillas seam side-down until grill marks appear, about 2 minutes. Cut each tortilla into 4 to 5 pieces.