I'm not familiar with the 2220, you might get more replies asking on the Microsoft Answers pages (this is where the old Nokia phones have their support now - I know, not obvious and there's a long story!).
There are two possibilities for the messages. One is that they're all stored on your SIM card, in which case when you put that SIM card in another phone the messages will be transferred as well. BUT, chances are any new phone (definitely iPhones) will take a smaller SIM card called a micro or nano SIM. This will be a new SIM card and you'll have to ask in a shop for them to transfer the messages from the old SIM to the new SIM.
The other possibility is that they're stored in the phone's memory, in which case they will stay in that phone's memory and you should be able to read them for as long as the phone works. It's possible you can transfer them as well but I don't know for sure. There was a software application called "Nokia Suite" which could be used with some older Nokias to backup messages, but I don't know if it worked with the 2220.
Which iPhone are you transferring to? Do you know that the new Nokia phones are every bit as capable as iPhones and much cheaper? The main differences are the aesthetic design and the software is different, but apart from that they're very similar and can use all of the most popular apps (except those exclusive to iPhone). The new Nokias are just as well built as the old ones (like a tank!), unlike iPhones seem to be and most of the new the models have guaranteed updates for two years. They are available in many countries now so you should be able to go to any local mobile phone shop to test a few out. If you're changing from a 2220 you will probably find one of the cheaper models (1, 2 or 3) totally adequate and also a reasonable size!
Whatever you decide, have fun with your purchase and good luck with your messages.
Thank you very much for replying. In the end I decided to experiment, so I took the old SIM out of the old phone, and switched it on, and found that it works fine without the SIM and the old messages are all there. Great! It seems odd that there's no way to transfer them, but maybe that's changed with the newer phones. Nokia Suite won't work with this old phone, in fact I could never get a cable to connect it to a computer anyway (and no bluetooth).
Anyway now I have an iPhone with a new SIM, and even found a way to transfer contacts using PhoneCopy.com .
Thanks again for your thoughts.
Glad to hear you were successful :)