Select the SAUTÉ function on the Instant Pot and set the heat to MORE. Add the oil and heat until the display says HOT.
Season the meat generously with salt and pepper. Add the meat to the oil and cook until browned, about 5-7 minutes. You may need to brown the beef in batches to get a good sear. If so, drain any accumulated grease between batches.
Add the onion, bell peppers and mushrooms to the pot and sauté until tender, about 3-5 minutes.
Add the garlic and sauté for 30 seconds.
Add the broth to the pot and scrape up any browned bits from the bottom of the pan.
Add the pepperoncini peppers, butter, and brown gravy mix. Stir to combine.
Turn the SAUTÉ function off. Secure and lock the lid.
Select MANUAL and set the timer for 15 minutes.
Allow the pressure to release naturally, then remove the lid.
Spoon out a few tablespoons of the gravy liquid from the pot into a small mixing bowl. Whisk the cornstarch into the liquid until it's smooth. Stir the cornstarch mixture into the beef tips and gravy.
Turn the SAUTÉ function on again, but this time reduce the heat to LESS.
Allow the gravy to come to a simmer and continue to simmer until the gravy begins to thicken.
Turn the heat off and allow the beef tips to sit for 10 minutes before serving.
Remove the pepperoncini peppers before serving. Discard the peppers or remove the stems, chop them up and add them back into the pot.
Season the beef tips with salt and pepper to taste. Serve over hot egg noodles, rice or mashed potatoes.