I do wash my braids. I have human hair extensions. The only advantage I can think of for not washing your braids is that they might stay as neat as when you had them done. But I love washing my hair. And if I was a little more organized I'd probably wash it more often than twice a week. So I don't mind that sacrifice. Although I am also quite a cheat coz I do my own braids so it's easy to repair any unsightly ones after the wash.
When you wash human hair extensions, they do get slightly loose. If you do not ruffle them all over the place they shouldn't unravel too badly. Sometimes the braided part starts to look less tight after a wash and the extension hair may slide down the braid and even fall off if your hair is short.
Perhaps a good suggestion is to put rubberbands at the end of your braids. Say, five at a time. You can use rubberbands coz it's not your hair you'll be putting them on. I'm hoping you've got quite a big part of the extension beyond your hair braided. So the rubberband is to keep the braided part braided. In other words, you put it at the end of the braided part. Make sure it's tight enough not to fall off during the wash. Then as when you shampoo, use a squeezing motion keeping your hair straight. Use the same motion when you run water through for rinsing. When you apply conditioner to your hair, squeeze again so it soaks into the hair well. (I usually then put on a cap and go under a dryer for 15-20 mins.) Then rinse for quite a while to get the conditioner out. But I have read of people who like to leave a bit of conditioner in their hair, so perhaps you could use a moisturizing conditioner or one like Nexxus'...is it Humectress?... which doubles as a leave-in and so might be safe to leave in your hair. And then if you haven't ever done so, practice putting in extensions on a sibling or even a doll. It's such a blessing to be able to do them yourself. Then you can always redo any that look bad after a wash.
But I believe there are people who don't wash their braids. I think I read of someone using witch hazel on the scalp and... I forget how they kept the hair clean. Maybe she will post again or someone could give the link to the post, and you can see if you like that better. Good luck in finding what works for you.