They do generally change, although not always, and the extent to which they change varies between servers. There are also potentially records of this as well. They will usually become useless eventually, however. Here's how it works:
Your ISP (internet service provider) owns a range of IP addresses, and assigns them to clients, including you. They can more or less assign IP's as they want. In some cases, an IP is "static" meaning it belongs to the same client for a long time, and some are "dynamic", meaning that they change. Bigger organizations may have a static IP's, whereas other home users often don't.
In some cases, you will retain the same IP as long as you are connected to the internet. Often this connection remains open if your modem is still turned on even if all the computers are turned off, so to completely disconnect, you may have to turn off the modem (I'd pull it out of its socket although you don't need to do this) and even then you may have to leave it unplugged for a few hours if not a bit longer, before the ISP recognises that IP as no longer in use and reassigns it. It's been a few years since I've ever checked mine, but I usually found that if I pulled it out and left it overnight, it would have changed by the next day. Alternatively, if we went overseas for a few weeks, it was always new by the time we got back.
So that's the first bit. The next part is whether or not the ISP keeps any logs of what your IP actually was. Your ISP may not actually retain this information for very long either- they usually retain it a bit for billing purposes, but often they may not hold it that long afterwards. Some countries now have laws on this, though- increasingly, many countries are requiring ISP's keep a record of which IP's were given to which people, often for about 2 years or so, but check local laws on this.
There are also various tricks with VPN's or proxies or whatever to hide your IP, which is a whole other story. And furthermore, people frequently get very concerned about IP's, who can trace them, how to hide them, and so forth, but often they aren't the issue at hand. There are two reasons why:
* ISP's will usually only give this information out with a court order or to law enforcement, which usually requires fairly serious matters for them to get involved. Beyond that, nobody can legally get this information most of the time.
* In many cases, good old-fashioned detective work is much more important and often more effective than simply IP tracing. No amount of IP hiding will protect you from this if you don't know what you're doing.
There was once some guy from Harvard or MIT or one of those places, who thought he'd send a bomb threat to the college to get out of an exam (to this day I wonder how anyone who thought this was a remotely good idea got into a top-level college in the first place, but anyway). He thought he'd covered all his tracks real well- he'd hidden his IP behind a VPN, and then he used some special email service that concealed his email address and the (already fake) IP he was using. Real slick, right?
Not so much- the FBI, upon reading the threats, realised pretty quick that the bomber was almost certainly someone from the university in question, and lo and behold, only one student from the university was using that VPN at the time the threats were sent. Busted.
To be fair, even then, if he'd kept his mouth shut and said, well, thousands of people use that service, it's probably a coincidence, he might have gotten away with it, but he was so shocked they had found him he gave it up. Which goes to show- you need more than just IT skills to deal with the cops.
So basically- there's no hard-and-fast answer on how long IP addresses change. The information will, in almost all cases, go out of date eventually, but it takes time, and furthermore, in all but a few circumstances, it doesn't matter.