Heat 2 tablespoons oil in a multi-functional pressure cooker (such as Instant Pot(R)) set on Saute mode. Add beef; saute until browned on the outside, about 6 minutes. Remove from pot.
Heat remaining olive oil in the pressure cooker. Add onion, carrots, celery, and garlic. Saute until vegetables are softened, scraping the browned bits into the mixture. Turn off Saute mode.
Return beef and drippings to the pressure cooker. Add tomato sauce, broth, spinach, wine, pasta, green beans, basil, and oregano. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 6 minutes. Allow about 8 minutes for pressure to build.
When timer is done, let minestrone sit, 1 to 2 minutes. Release pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove lid. Add black beans; season with salt and pepper.