Turn on a multi-functional pressure cooker (such as Instant Pot) and select Saute function. Add sausage and sauté, breaking it into pieces with a wooden spoon, until browned, about 5 minutes. Add water, chicken soup base, beans, zucchini, onion, carrots, and garlic. Stir once.
Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 3 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 spinach, Italian herbs, and rosemary; stir until spinach has wilted. Ladle minestrone into bowls and top with freshly grated Parmesan cheese.
