You do not have to but it is recommended to promote growth. When you remove the old stem and or flower, it creates new flower growth faster. In the case of bulbs you don't want to remove the greenery till it has browned. The reason is, the nutrients from the leaves goes back down into the plant, causing the bulb to multiply.
For continuous blooms you need to deadhead perennials and annuals. The flower is its way to reproduce. It takes a lot of energy for the plant to produce seeds from the flowers. Often, once the plant has made seeds it has done its job to reproduce and starts to decline and/or die off. When you deadhead, it thinks that it can't reproduce because there is no flower to make seeds. Voila, it sends up more flowers to do its job, reproduce. For many plants, the more flowers you cut, the more you will get. Rudbeckia is a tough plant, especially with a bit of neglect and the gold finches love its seed heads. Coreopsis is a bit pickier and really needs deadheading all the way down the stem. Enjoy the flowers!