For the sake of convenience, this happens on Earth. However, in this scenario only the American continent (all of it) is inhabited.
One of the countries is experimenting with extremely advanced and extremely hazardous power generation technology, codename project Nova. The Nova facility is located underground somewhere below the Amazonian forest. You can move the facility somewhere else if you need to, just keep it far from any population center. Obviously, the experiment goes horribly wrong.
The experimental reactor ends its life in a blaze of glory, that is to say kaboom. The explosion triggers the biggest earthquake in recorded history. In the wake of the quake and aftershock, there is quite a bit of destruction.
On the surface the damage is extensive but it's nothing that can't be repaired/rebuilt. Below the surface, the quake has shaken the ground so hard the planet will become uninhabitable. When will it happen? The continent will be come uninhabitable in 50 to 100 years, the rest of the planet will eventually be as inhospitable. On the bright side, everybody will get evacuated, so hooray for FTL and unlimited budget.
Anyways, I need a good reason why the planet will become practically uninhabitable.
My thought was that the quake would trigger a chain reaction that would result in increasing seismic and volcanic activity that would eventually shake every building down, block the sun with clouds of dust, and flood the streets with lava. Is that even possible? If not, how can I effectively end the world starting with one big explosion?
I'll clarify on the requirement. The planet doesn't have to be actually uninhabitable. If there's enough casualties and the living conditions become too bad, governments will start looking at their options.
In short, there are three Earth-like planets at different places in the galaxy. FTL technology is limited to one FTL drive of alien origin and one reverse-engineered, almost complete prototype. Both are untested but the technology would be (successful, something has to go right from time to time,) used to evacuate people off. The process would take decades, but would be over before living conditions became too insane.
That option has to be more appealing than whatever will happen if they stay. They don't have the technology to terraform, nor the desire to.