I've done it two days in a row a couple of times and didn't suffer any bad effects, but I definitely wouldn't do it several days in a row. The reason the workout tells you to take a day or two off is to give your body a chance to recover to the point where you can push yourself just as hard next time as you did this time. If you think you can do that the day after, by all means go for it.

Just be sure to pay close attention to your body and what it tells you. Pushing yourself is a good thing. Wearing yourself out is not.

For beginner runners, I'd be far more worried about the impact you're causing to tendons and ligaments that aren't used to it. If you start to have pain in your shins, joints, calves or feet then you should look to take some recovery time rather than trying to work through the pain.
There's a big difference between training your muscles and cardiovascular systems and your joints and tendons recovering from injury.

Common wisdom says 1) listen to your body and 2) new runners should rest at least one day between runs. If you're new to running, consider running every other day to help prevent injury that might sideline you completely for a while.

Walking definitely isn't a problem. There's (at least) two reasons for beginners to take 'rest' days.

1. You build muscle while resting. During exercise you get micro tears in your muscles. That's a sign for your body to make your muscles stronger. But it takes some time to do so. So, if you do not get enough rest you're creating new tears before the old ones have healed and in that way stop your progress.
2. Your tendons, ligaments and bones get stronger slowly, slower than your stamina increases. But running does put a lot of tension on them, so by taking rest days you also spread out your training over a longer period giving your body time to strengthen your tendons, ligaments and bones.

This also makes it clear what you can, and what you shouldn't do on your rest days. You're better of not training your leg muscles heavily because they need the rest. You can do any exercise like walking that doesn't tax your legs very much. You can train your core, your upperbody all you like.

Now, for non-beginners this is different. Firstly there tendons and such are already stronger, so they do not have to take it slowly to build their strength over time. Secondly because their muscles are stronger already they can also run more often. Unless it's a very long, very fast, or otherwise taxing training the will not get as many micro tears. If you look at professional athletes their training schedules can easily be two trainings a week, but the trainings themselves will be vary.

I'm a beginner, just getting back into exercising regularly (let alone being overweight). I'm just starting week 2, but I definitely take my rest days. I just exercise differently on my off days. I do light stretching exercises for my legs, upper body, core, and some light weights (because I'm a wimp atm). I don't give up on the exercise, so I don't loose my motivation either, I just refocus the part of my body.
Yes, if you feel comfortable running, personally I don't think there's anything wrong with it, if you're just starting out or feel uncomfortable, then stop. Pushing yourself just to keep motivation is not a good reason.
dragon1ady and MissMormie also had some solid advice.

Don't call them rest days.

Call them Repair Days.


They make sense then.

Doing yourself permanent damage rather than say 'I gotta take a break' isn't heroic. It's daft. I've got RA, so I know what joint problems feel like. I've also had most overuse injuries in my time because i was stupid and thought it was more important to ignore my body telling me to stop. If you do permanent danage to yourself through overtraining, you aren't going to like the result when it takes away your normal, healthy life, trust me.