Well, the dish soap never dried, so I tried the hair dryer. I had... mixed results. Fortunately, the soap is now dry enough that I feel safe shutting the book. Unfortunately, the soap is not exactly dry. It's still a little sticky, although I pressed another piece of paper on top of it to remove excess stickiness, and it's got this almost waxy feel. It'll probably stick to the previous page. It's also got all kinds of dust particles stuck in it from when it was sitting open. Yaaaaay.
The book itself really suffered from the experience. Leaving the book sitting propped open flat for so long was bad for the binding, and I think the hair dryer actually melted the glue a little bit. This page has halfway detached from the binding. It's also got all kinds of waves and ripples in it from having liquids dry on it. Heh. Good thing this is Wreck This Journal and not Keep This Journal In Perfect Condition.
So if any of you were considering using liquid dish soap as a substance sample for your own Wreck This Journal, my recommendation is: don't. In fact, now I'm afraid to try any other kind of soap or sticky thing. I had really good experiences with very watery liquids, such as perfume and hand sanitizer, and things that were meant to eventually dry, like nail polish. Paint would obviously work quite well, and I'm sure food coloring would be a good choice. But not soap. Maybe shampoo? I know that, if you don't rinse shampoo out of your hair, sometimes it can become dry and flaky (and itchy). But I'm not about to try it. If I do continue with this page (and after my experience, I might just leave it the way it is), I'm going to steer very clear of any kind of soap and anything with glycerin.