Will missing birth control pills one month cause side effects the next month? such as breast heaviness but no soreness?

No: Missing pills one month doesn't have an effect on the following month. Missing pills can change when you have your withdrawal bleeding (menstrual cycle), but after your cycle is complete you don't have to worry about any side effects. Breast heaviness can be side effect of pill, generally, can also be a sign of water retention or weight gain.