Generally, the animation functions of these programs are separated from the programming functions. To use Flash as an example, you can draw, color, create keyframes, add shape and motion tweens, manipulate artwork, create new scenes, create interactive buttons, and do numerous other things without doing a single jot of programming. The only technical skills you need to learn are how to move a mouse, how to use a graphics tablet, and what buttons activate which tools.
But to again use Flash as an example, you can also greatly enrich your animation skills and your final product by learning and integrating programming tools. With ActionScripting you can control how users interact with your animation by triggering events based on mouse or keyboard activity; you can control playback between scenes and keyframes, allow users to navigate information or view different animations based on choices they select, even control the type and rate of motion for various objects. Integrating HTML, CSS, and XHTML opens up an entirely new suite of web interactivity, allowing the content of the Flash movie to branch out further in how it integrates with both internal and external web content. PHP lets you capture and pass variables between different pages and different animations. The possibilities are endless, and as varied as your imagination and your willingness to learn.
The question really isn't if you have to learn computer programming, but if you should. If you're part of an animation team, often you won't need to. There'll be someone to handle the programming, and while you may need to learn one or two scripts or commands to finish off your part of the project, for the most part it'll be in someone else's hands. If you're working independently, as a freelancer, or as part of a very small group, though, you may want to pick up a little scripting know-how.
One problem can be dividing your time between honing your animation skills and honing your programming skills, especially if you aren't technologically inclined, resulting in a difficult learning curve. My suggestion is to pick the programming language that you'll be using the most often in your work. Pick up a few books on it; look up some online tutorials. Find sample scripts and break down how they work. Practice with simple commands first, to see what they do and how they affect your digital animations. It's often easier to learn in practical contexts, when you have a specific purpose and you need to learn how to accomplish that purpose.
If nothing else, learning these skills will make you more comfortable in a digital environment, more conversant with others of your craft, and more viable when seeking new work. And who knows - once you learn your first programming language, you might find you like it and turn into an animation coding dynamo.