Can you legally stream TV shows online as long as you don't download them?

This question is unclear. It is hard to tell what it is you are asking. You mention mp3s. Those are music files and would be songs, not TV shows. Youtube has video files, not music files. If a TV show has been legally placed online for you to view, then you may view it for your personal viewing enjoyment, but you may not display it for public viewing or charge money. There are many full episode shows that are online legally and that are legal for you to watch. Are you asking if a television show full episode has been illegally placed online, if it is legal for you to view it, as long as you do not download it? It is never a good idea to knowingly participate in someone else's illegal activity because it may also be illegal for you. To my knowledge, it is not a violation of the U.S. Copyright law to view a program that has been placed illegally online by someone else. If you are knowingly seeking out such illegal materials or paying to view them or paying to belong to have service that provides them, you might be breaking some other law of the nation or state or city where you are doing this. If you are doing this at your job or school or library or using a job or school or library computer, you might be breaking their rules.

You can't legally stream or download someone else's program unless the owner of the copyright to that program has given permission for the show to be streamed and/or downloaded. Copyright law applies in both cases.