When you're significantly overweight, is it safe to under-eat and over-exercise every now and then?

No. You risk injury. Being obese also predisposes a person to back and knee injuries, which could recur and become life-long issues. To get any decent result, it is very important to make gradual changes to diet and exercise, in a way that you can stick to consistently. Unlike laundry, you can't "save up" exercise and do it all at a later date.
Not over-exercising. An obese person who "under-eats" occasionally should not have a problem, but will be very hungry that day. If he "over-exercises" occasionally, he will have problems due to being extremely out of shape. One should choose a daily exercise he really enjoys. Any form of exercise burns calories, so if one does not eat or drink more calories than he is using during his exercise, he should lose weight.