Melt butter with olive oil in a large heavy pot or Dutch oven over medium heat.
Place flour in a shallow bowl and dredge the beef tips in the flour. Brown the beef in the hot butter and oil, stirring often, about 15 minutes.
Pour water into the pot and scrape to dissolve browned food bits from the bottom of the pot. Stir in the flavor packets from the rice mix and dissolve; mix in the rice and mushrooms. Bring the mixture to a simmer and reduce heat to medium-low; simmer until beef is tender and the rice has absorbed the liquid, 20 to 30 more minutes.
