I ride my bike to work Monday through Friday, about forty minutes both ways. The elevation changes a bit but not too much. I'm not a huge fan of holding the handlebars, and I often ride with no hands for as much of the distance as I can. I know this throws off my balance a bit (and a large bump is more likely to throw me off my bike) but beyond that should I be worried about how this affects my muscles? E.g. could I be straining them/stretching them in the wrong way? In general is it bad to consistently ride using that posture?
Of course, you will have less control over your bike if you don't hold the handle bars, especially if something unexpected happens (like a truck coming out of nowhere at full speed for example).
Regarding sitting upright, you are very lucky to be able to keep your back straight like that, most people have a bad posture, and are not comfortable sitting upright on bar stools for example (or any chair without something to hold your back). Being upright like that uses your lower back muscles and your core more generally, which is a very good thing. Also you will relieve potential stress in your forearms, because they are not resting on the bar (with your weight).
I put my road bike on a trainer, raised up my desk at work, and decided I could casually ride all day long while typing away at my desk. At first, it was a lot of fun and I was effortlessly logging a good 20 miles a day. However, after the first couple days my tail bone started to hurt a lot. I got a softer seat, and it continued to hurt. I'm no doctor, but I'm pretty sure I bruised my tail bone due to extended up-right sitting on a hard road bike seat. I couldn't comfortably sit on anything for a month!
Let my mistake be a lesson!
I do that a lot and haven't had any issues.
What I have noticed is that sitting back at times gives your legs a rest from other postures and helps you flush the pain out while you can still maintain a decentish forward momentum. You can also twist, turn and stretch your arms, shoulders and back.
The downside (apart from balance) is that you can't get the same amount of power into the pedals since you are less stable and don't have anything to push/pull against with the rest of your body.
But you're not racing anyone in that posture so it doesn't matter.