Alright, I admit that title's an overstatement. You'd probably have to do a lot more to lose your job than not keep a to-do list.
But pretending you can keep all your projects organized in your head can be a simple start to a downhill slide in the workplace. In an environment with ongoing projects and constantly shifting priorities, it's tough to keep track of what really needs to get done and what can wait. It's even tougher when your home life is just as busy as work, and you have trouble finding that ideal work/life balance we so often preach about.
So unless you have a sharp, never-fail memory, you're bound to forget a task or two if you don't write them down. Nothing is more damaging to your credibility in the workplace than having superiors ask for updates on projects that you forgot about. ("Oh, that project. I, uh, well, umm...") And it can be even worse in the job search. Did you ever show up for an interview without some key piece of information they specifically asked you to bring? A simple to-do list can save you from these professional crises.
My favorite (though not most secure) method for keeping track of my projects is a simple Post-It note stuck to my monitor. Not only is it right in my face all day, but I get a sense of accomplishment when I've crossed off enough items to create a new list that reflects all my new priorities. Less colorful but still effective methods include the traditional handwritten list on a lined yellow tablet; the Word document; the elaborate spreadsheet; the erasable whiteboard list; or the latest iPhone app for project management.
And if anyone is curious about what you're working on, they can just glance over at your to-do list and get a brief idea. This is also handy when your superiors ask for that update. If it's on your list, you don't have to admit you forgot about it—you can claim you simply haven't gotten around to it yet!