No, they won't. Granted, cartoons on the big networks are going the way of the do-do, but there's still 24 hour animation channels like Cartoon Network, kids' channels that prominently feature animation like Nickelodeon, On Demand, the internet and DVDs. Cartoons will be just fine.
Not trying to be a jerk here, but quite frankly, I'm getting just a little bit tired of people moaning about these program blocks that are over 5 years old going away. Toonami was on CN for 11 years. ELEVEN. That's a long time for a program block to be around. People also tend to forget that Toonami was around a couple of years before it died. Toonami's lineup got bad the last couple of years, which is what prompted CN to drop it.
Also, I'm tired of people berating the networks who make these decisions by claiming that they are "taking away their childhoods" and calling them "mean" for taking their favorite shows and blocks away from them. "Mean"? What are we, in elementary school? The networks couldn't care less about your feelings or about your childhoods. Like Silverstar already said; entertainment is a business, like any other, and the decisions that they make are business decisions. If you run a company, and someone on your staff routinely doesn't live up to expectations, then you have no choice but to fire that employee. It's the same way with network programs and programming blocks; these blocks were no longer the big ratings grabbers that they once were, and you're no longer pulling your weight, you've got to go. Period. that's all there is to it. It's business. Never personal.