I'm a little confused about the sets/reps that you've laid out here. I *think* what you're saying is that you do the burpees, dips, and so on for 12 reps each with no rest in between (like one big set -- called a super set), then do that super set 4 times. If so, this is a pretty effective and somewhat intense routine, especially if you are still a beginner. I would strongly recommend a rest day in between, with two considerations:
Consideration #1.) What form does the "cardio" element of your workout look like? You list the treadmill and mention that you've done some HIIT training in the past, but how are you working it into this routine? If the sets/reps are performed as I described above, the "strength" part of this workout might also give you some cardiovascular benefits. If you are doing a more traditional cardio workout on top of this (like on off days or during the "treadmill" portion), then I think a rest day is definitely in order. Look at the balance of "cardio" to "strength" training in your workouts and strive to make them at least equal. If you want to be really aggressive about losing weight, focus more on cardio and body weight training rather than lifting heavy weights. Lifting and building muscle definitely has its place in a weight loss program, but if you are a beginner it is best to focus on getting your nutrition on track, getting tot he gym, and getting your heart rate up while you're there.
Consideration #2.) Would taking a rest day discourage other physical activity on that day or the following day? For example, sometimes when I take a rest day, I end up not working out for two days because I've let myself get out of my routine and let other things in life take over. With the goal of losing weight, you should be hitting the gym (or doing some other exercise) as often as possible, preferably for at least an hour a day. If you take a rest day from this routine and do something else (like play basketball, swim, or go to a spin class), will you be likely to get back to it the following day? Make sure you work exercise into your routine first, then try to fine tune.
So, all that being said, I think the bottom line is getting in the most activity your schedule can afford without injuring yourself. It is probably not dangerous to do this routine without a rest day, but performing it three days in a row will not confer any special benefits either. In my opinion, nutrition is the number one factor in weight loss, the exercise. When it comes to your time at the gym, strive for quantity (within reason) and quality first, then variety, then all the other details.
Sorry for the long rambling post -- hope this gives you some food for thought. Keep on rocking on!