While it’s not mandatory to carry your Tourist Card with you at all times, it’s mandatory for exit.
Many travelers choose to make a copy of their Cuba Tourist Card so that they will have it for various situations, such as checking into accommodations or dealing with authorities. Keep a copy of the front page of your passport with you as well.
By carrying a copy of your important documents, you will always have them with you and can secure the originals in a protected place, such as in a locked drawer or safe at your accommodations.