Something would definitely happen—but it wouldn't be a big something.
As mass moves towards the center of a rotating object, the rate of rotation increases. Imagine, for example, an ice skater pulling her limbs in to increase the speed of spinning. So, technically, you can make the Earth rotate more rapidly all on your own simply by crouching down on the ground—but at a totally negligible amount.
Powerful earthquakes can redistribute mass in a slighter more impactful—but still ultimately negligible—way. The earthquake in Japan in 2011 moved so much mass toward Earth's center that every day since has been 0.0000018 seconds shorter. However, if we tried to recreate the force of that earthquake simply by jumping, we'd would need seven million times more people than currently live on Earth.
But what about all of us in one place pushing down on the earth as we jump away from it? If everyone in the world stood shoulder-to-shoulder, we could all fit into a space the size of Los Angeles—that’s more than seven billion people packed into 500 square miles. But even if we all jumped at once in that small area, not much would happen. Our collective mass is an awful lot—just not compared to the mass of the Earth.
Let’s assume we all jump 30 centimeters. Our force would propel Earth away from us, but not very far at all. Earth would only move away from us 1/100 of the width of a single hydrogen atom. Additionally, just as we would return to our original starting position on Earth, the Earth would return to its original position in the universe. No lasting effects here.
Over on YouTube, VSauce covered this topic in more detail. Watch his explanation: