Just to agree that being connected to the Internet doesn't use up bandwidth allowance. Leaving a web/forum page open doesn't either unless there is (as mentioned) streaming video or other moving page content present. Uploading, perhaps a photo or video, will use up bandwidth allowance too.
Going a web/forum the page in the first place uses bandwidth because the page content has to be downloaded for displaying on the computer. Data transfer is occurring at that time and for that read 'bandwidth usage'.
Having said that, there is a factor that saves bandwidth usage. Browsers have a cache. When you visit a site, a lot of its page content is downloaded to the computer and stored in the browser's cache. When revisiting that site/forum/webpage, some of the content will drawn from the browser cache and isn't downloaded again. This saves bandwidth usage so won't use up the allowance. This certainly applies to images although (I believe) not videos.