I've found a way to practice Danish on mobile, right now!
TL;DR: Open the app, and click the "Practice Weak Skills" button at the bottom. Don't be discouraged if things don't look right, you'll be presented with Danish practice. If you have a warning pane slide up from the bottom telling you "Unsupported Language", exit the app, start it again, and be quick to hit the blue "Practice Weak Skills" button, and it will work.
I opened up the Duolingo app on my iOS device (probably Android works the same way, but leave a comment if not), and wandered around a bit. Pretty much everything looked like I left it before I started learning Danish-from-English on the site. Namely, it looked as if I was still learning English-from-Hungarian: The skill tree, the flags, etc... But as I wandered around, I noticed some signs that the mobile app does know that I'm learning Danish: there were some mentions of "da" and "da||language with article" like the blanks are not filled in.
When I clicked the big blue "Practice Weak Skills" button at the bottom, to my surprise, Danish practice was presented to me.
Then I dropped everything to write this post because I know many are eager to learn Danish (or other "unsupported" languages) on mobile.
Edit: Screenshots, or it didn't happen: http://imgur.com/a/f5Rqq
Edit 2: The second time I tried to reproduce, a warning slides up from the button telling me "Unsupported Language" pair, "da from English". That made me very sad. Why would the devs block people from accessing something that essentially works already? :(
Anyways, I found, that if I am quick before that warning pane comes up, I can still hit the "Practice Weak Skills" button, and that would prevent the warning to slide up, and I could still practice Danish. So that made me happy that it is not entirely blocked, and with enough determination, we can get to what we want. :)
Edit 3: It is also possible to learn new skills, not just practice old ones! But the experience is a little more flakey. Here is what I found:
I noticed, during my consecutive starts of the application, that it was no longer the English-from-Hungarian tree that I saw at startup, but rather the Danish one, in its full glory, showing my current progress, too. So I wanted to get at it somehow. I tried to tap a skill, then a lesson, but no matter how quick I tried to tap, the "Unsupported Language" warning pane always blocked me off. :( Then I got an idea: What if I start a "Practice Weak Skills" session, just to discourage that rude pane from blocking me, then I quit that session, and then continue? And to my surprise, that worked just as expected: I had full control of the app after that. So I went on to learn about Danish colours, such as how "en appelsin er ikke en farve".
The full lesson worked just as expected, without any hiccups, except when I got to the very end. Instead of getting bells and whistles for my accomplishment, the rude pane blocked me just as I finished. I got worried that my progress was lost, but when I went on the website, I saw that I am 11 xp higher, and Colors-2 is also marked as finished. I think I can live with that tradeoff.
So this morning I am a very happy man that both of these most common use-cases are possible to achieve even when I am on the go. :)
Edit 4: A word of warning: I found that sometimes when I "context switch" out of the mobile app to another one, or just pull down the iOS notification-pane from the top, or the quick-settings-pane up from the bottom, or just simply lock the phone, when moving my focus back to the Duo app, the rude "Unsupported Language" pane rears its ugly head up, even if I was in the middle of a lesson or practice.
Once that happens, there is no way of getting rid of the pane, other than forcefully exiting the app, and losing progress. So try to do one session in one sitting, otherwise you might lose some of your progress.