Sudden speedup implies sudden catastrophe.
Lets assume a sudden doubling of rotational speed. You suddenly have 1000 miles per hour winds at the equator at the atmosphere is now moving far too slow to keep up. Same problem with the oceans no longer matching the nearby Earth.
Energy required to perform this speedup is beyond incredible. Earth has a rotational energy of about 2.138e29 Joules, to double the speed would require 6.414e29 J - this is about 65% of the binding energy of the planet Mercury, a bit more than 100,000 years worth of solar energy hitting the earth. Even at 99% efficiency, the waste heat will cook the Earth far beyond survivability.
OK, hand of God time (reverse Joshua effect). Still quite a bit of destruction as the new speed requires the earth to become more oblate., i.e., the polar diameter will decrease and the equatorial diameter will increase. Major earthquakes and volcanic eruptions globally, climate impact will be huge (nuclear winter times 100 perhaps). It will take many millennia to reach a new balance. Still, life should not be completely extinguished and eventually you reach a new equilibrium.
At higher speeds, you would have more transitional problem, but you start having even bigger problem such as notably reduced atmospheric pressure at the poles. And the earth start losing atmosphere at a more rapid rate. Just how fast you all die off depends on the new speed.
If you reach 10 times the rotational speed, you are all dead fairly quickly as the atmosphere would be lost pretty quickly, followed by the oceans boiling off. Not instantaneous death, over the course of a normal lifespan Earth would become mostly dead.
At 16 times the rotational speed, the atmosphere would be gone in minutes as the rotational velocity has exceeded the speed needed to enter orbit, i.e., things on the surface near the equator would simply start orbiting earth.