It's not always sunny and 73° where I live. Is there any Weather forecasting app for iOS which will change its icon to show current conditions? Or even a way to make the standard Weather.app behave like this?
There are two apps, Celsius and Fahrenheit, that I know of that will add badges to their icons to attempt to give you an up-to-date temperature at a glance. I say attempt because the badge approach doesn't support negative numbers so if your temperature dips below zero for either of those two scales you won't know.
But without jailbreaking your phone that's as good as it gets. The App guidelines from Apple don't let third party apps change their icons.
You're looking for Pocket Weather World or Pocket Weather World HD by awesome Australian developers Shifty Jelly. It will display the current temperature (for your most recently-viewed location) as a badge on the icon.
If you're in Australia look for the local versions (i.e. Pocket Weather AU HD which offer better integration with the Bureau of Meteorology's radar services etc.
(iOS icons can't change dynamically, the badge is the best they can do so far)