Not if you're the developer of the built-in weather app.

Since the app bundle is signed, and the icon is inside the app bundle... no, but Apple can cheat for their own apps: Calendar's icon always shows today's date. Clock and Weather could update continuously too. For third party apps, I'd like my stopwatch to keep updating it's icon in the background, and one of my apps could do with it too.

A more serious problem is not being able to update Default.png, so if your app loads the last view the user had, it's hard to make it seamless.