Turn on a multi-functional pressure cooker (such as Instant Pot) and select Saute function. Add beef, mushrooms, onion, garlic, salt, and pepper. Cook and stir until browned and crumbly, 5 to 7 minutes. Drain and discard fat.

Add tomato sauce, diced tomatoes, water, dry noodles, soy sauce, Italian seasoning, and bay leaves. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 4 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. Remove bay leaves before serving.