Turn on a multi-functional pressure cooker (such as Instant Pot(R)) and select Saute function. Add ground turkey and turkey sausage. Cook and stir until browned and crumbly, 5 to 7 minutes. Add tomatoes and tomato soup. Close and lock the lid.
Select high pressure according to manufacturer's instructions; set timer for 8 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid. Add chili seasoning, garlic powder, cumin, and smoked paprika. Stir to combine.