Student posts and comments can be removed by changing their status to "Trash".
This is most easily done using the bulk editing feature available in "List" view. Navigate to your class homepage and click the "list view" icon at the top right (above the search box). Next, click the checkbox next to the posts or comments you wish to trash and click the blue "Edit" button. Posts and comments will be removed entirely from the class when changing their status to trash.
You may also change the status to "Trash" when viewing each individual post.