Combine chicken broth and orzo in a multi-functional pressure cooker (such as Instant Pot); stir to combine. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 5 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. Switch to Sauté function. Stir in butter, garlic, spinach, and mushrooms. Season with salt and pepper.