Actually, wouldn't the change in the Earth's velocity instantly crush a portion of people into the ground on the wrong side of the planet and fling people high into the atmosphere on the opposite corresponding side? Everyone would die from that alone.
There would not be any net force change on the earth as a whole, if that is what you are implying. Our earth circling the sun is basically the same as the earth constantly free falling and all of the people are freefalling with the earth as well. There is no real force being applied. Therefore, if the sun disappears, there is still no force.
However, the tidal difference in gravity from one side of the earth to the other would disappear. Would this cause any major earthquakes? I don't know. The earth is making one full rotation every 24 hours, so those tidal forces are rotating every 24 hours around the entire earth anyways, pushing and pulling at it from different angles. However, if that tidal force was instantly turned off it might give a little earthquake.