First, it's worth noting that getting your IP won't identify your wallet address. So they can't connect to you and then look you up on the blockchain to see who you've been trading with. Associating an address with a person takes normal investigation, like searching your computer or associating you/your IP with a post on the internet containing your address.
Unless someone logging the connections to their bitcoin client and they decide to make it public, there isn't any public record of it.
That said, what most would be worried about isn't that their IP would be publicly published, but that certain organizations may be able to identify them. One way these organizations could identify you is by running their own bitcoin client and logging their internet traffic (TCP on the port BTC uses). This would be very easy to do, so if it is something you're concerned about, you should assume it is being done. (If you want to know how easy, look up wireshark. It can log all traffic on your local network.)
Assuming someone is logging IPs on the bitcoin network, the only hope of not being logged is that a singe node only connects (on average) to around 8 peers (but this is configurable). This small number means you have a somewhat small chance of connecting to them.
But, depending on how peer discovery works, it may also be possible to be found by the logger asking for peers. That is, there has to be some way of finding peers in the btc network. This could possibly be exploited to identify a large number of IPs connecting to the btc network. (To be explicit, I haven't studied the specific way peer discovery works, but from general understanding of programming and networks, it seems very plausible.) If this were the case, then the fact that a node only connects to about 8 peers is pointless, because you wouldn't need to connect directly to the organization to get logged; they could find you by asking others for new peers.
If you send them to a paper wallet and never connect to the btc network through a client running on your computer, then no, your IP can't be discovered. But, the only way to retrieve them from the paper wallet is to import it into an digital wallet of some kind, which, when you connect that digital wallet to the network to spend them, will identify you.