Another user could 'fake' their IP address information.
This is very easy to do - see: http://www.makeuseof.com/tag/how-to-mask-yourself-online-use-a-fake-ip-address/
The presence or absence of log files on the PC will help prove a case.
There are many types of logs that can be set up on your computer.
For example, when a new IP is obtained it will show up in the Event Viewer.
If you have an internet log set up, it will show every site you accessed and when.
Some routers will also log IP activity through them.
You can print out a log and forward to the site moderators, however the logs can also be manipulated by someone who knows how to do this, so you should check first with the site's moderators what type of logs they might accept as proof that you did not make the posts.