A Safe House will upgrade when the required number of Survivors are dropped off at it. You can see how many this will be on the Safe House's information screen which appears when you tap on it.
The evacuation timer, which tells you the time until a Safe House will disappear starts as soon as the first Survivor is safe inside and will reset each time it upgrades. If the required number of Survivors is not met by the end of the 10 day timer then the Safe House will disappear.
Upgrade Safe Houses to get more rewards each time a Survivor is added and to stop them from evacuating!