I assume that someone may have tried to upload a Bitcoin app and it was rejected, but does anyone have information on what specific rules Bitcoin-related applications violate on the App Store?
As far as I know, there have been no submissions of Bitcoin apps to the Mac App Store.
A Bitcoin app for iOS was submitted to the App Store in May 2011 by Ehud Ben-Reiven. He reported that it was rejected because "...it involves intermediate currency...". He subsequently open-sourced the code, bitcoinApp, but, as of this writing, it hasn't been updated in years.
Blockchain.info had some success in submitting an Blockchain app which uses the site's API to do all the heavy work. However, it was removed without reason in May 2012 and later restored in August 2012. This app relies on Blockchain.info as a hosted wallet, though, and is not a fat client.
TL;DR Apple disallows Bitcoin clients because it considers Bitcoin to be an "intermediate currency".
Speculative TL;DR Apple disallows Bitcoin clients directly because it's a payments system that circumvents Apple's desire to take a cut of all money that flows within its app ecosystem. It's not known how Venmo and other payment networks are allowed to circumvent that, if they even do.