Well ... hmm. I think there are a lot of variables here. First of all are you talking about the bare minimum you need to be healthy? To build cardio-vascular health? To lose weight? To lose weight w/out losing muscle mass? All of those require different levels and different types of exercise.
Then there's the personal part of it - each of us is different in weight, strength, shape, and condition. 45 minutes for one person might be as easy as breathing, while 45 mins for another person might be enough to leave them a sweating, puking pile on the floor.
Ok that said, then there's the whole "lose weight faster" thing ... time on the treadmill really burns very few calories, comparatively. Obviously this varies from person to person, but in extremely general terms, an hour of moderate to moderate-high intensity time on the treadmill will burn 400 calories. So putting in an extra 15 mins a day would equal an extra 100 calories a day. That's 700 calories a week, if you do it every day. That's about 1/5th of a pound a week.
Then, let's get into the idea that you really should be doing more than just running on the treadmill. If you're serious about losing weight healthily, you really should be adding in weight lifting to your routine. Weight lifting and body resistance exercises will minimize the loss of lean muscle mass and maximize fat loss. Then it becomes not "will I lose weight faster" but "will I change my body composition more effectively".
Ideally you would lift weights 2x - 3x per week and do some form of cardio 2x - 3x per week to maximize both body re-composition and fat burning.
Now, all of *that* said .. yes you do need to take a day off. Your body and your muscles and your nervous system need recovery time. Does it have to be 1 day a week? Maybe not. Some people do great on 1 out of 10. Some people only work out 3x a week. But if you don't take a break eventually you'll go backwards - and I'm not just talking about weight loss, but about overall ability. If your body doesn't have recovery time, you'll wind up not being able to effectively complete a workout. You might start to feel fatigued, stressed, etc. And eventually you'll make yourself ill by overworking your body.
See, it's not just as easy as saying: do x for y minutes and you'll get z result. It just doesn't work that way because we're all so different. There are basic fundamentals that hold true for everyone: If you want to maintain lean muscle mass you have to work the muscles (which means weights and/or body resistance). Cardio is good to aid in losing fat but also to build cardio vascular health. Rest periods are necessary. Eating healthy food in reasonable quantities is necessary. Within those guidelines - there's a huge wide range of what works and what doesn't work.