Based on how Bitcoin works, and your knowledge of the overall Bitcoin ecosystem, blockchain, addresses, wallets, hackers, etc ...
Is there any risk of entering your public addresses into a monitoring system, to make sure your balances are always there, and no coins are moving?
Simply put - its drawing attention to the existence of the address, whereas it may have never been seen on the network previously.
Then again, when you load BTC onto an address, it has been "seen" on the network.
But that doesn't mean the dude who made your iPhone app knew about it. Until you load it onto his app.
Now you've got an address with some serious money in it, and some dude knows its there.
Any risk?
The mere fact that an address has any funds in it at all means that a transaction involving that address has already transpired, and therefore is publicly viewable by anyone with a copy of the blockchain. Unless it's empty, then the network already knows that a particular address has funds in it. Monitoring your address is no different than telling it to another person in order to receive payment.
It's relatively easy to query the blockchain for any address containing over 1 Bitcoin. Anyone who wanted to find "an address with some serious money in it" would simply write code to do that instead of coding an address monitor and waiting for a random user with a large Bitcoin stash to use it.