Combine first 3 ingredients in a bowl. Melt butter in a large nonstick skillet over medium heat. Add mushrooms; sauté 10 minutes. Add peppers; sauté 1 minute. Pour egg mixture into pan; cook until bottom begins to set, stirring to scramble. Remove from heat; stir in cheese.
Warm tortillas according to package directions. Spoon one-fourth of egg mixture down center of each tortilla; roll up. Serve with salsa.