I found a great illustration on the Internet that I want to add to my online course. The person who posted it must know that it can be easily copied, so isn't that implied permission?

Publishing something on a web site is no different from publishing it in a more traditional format. The person who created it controls its duplication and distribution unless it is clearly stated that the material may be copied or is in the public domain. In addition, the person who posted it may not have created it and the posting itself may violate copyright.