Mix chicken, oregano, tapenade, feta cheese, and garlic together in a bowl; form into 4 patties. Season burgers with salt and pepper.

Preheat an outdoor grill for high heat and lightly oil the grate with olive oil.

Cook patties until charred on bottom, about 5 minutes. Flip the burgers and finish cooking, refraining from flattening with your spatula until no longer pink in the middle, about 5 minutes more. Drain on a plate lined with paper towels before serving.