The operating system doesn't have much to do with the rendering of web content. It definitely won't influence the page layout (i.e. the placement of blocks and the flow of text); that's the job of the browser's rendering engine. I know of no OS that explicitly pokes into browsers to change where stuff appears.
The OS can, however, affect smaller things like font rendering. Windows, for instance, has a feature called ClearType that has been known to impact Chrome's text appearance for the better. Operating systems without that feature will therefore see slightly different text, and different OSes will have slightly different font rendering anyway. Also, some OSes have convenient ways to change which font is used. Video drivers and the physical video card could affect the rendering of anything that the browser uses hardware acceleration for.
Additionally, "native" UI elements like check boxes, drop-downs, and text boxes will definitely have different appearances from OS to OS unless the browser explicitly does its own special rendering.
Besides those, you're not likely to see large differences between, say, Firefox on Windows and Firefox on Linux. If you do, there are almost certainly differences in the browser code that is used in its builds for the different OSes.