Which country has no permanent citizens by birth?

Vatican City. Citizenship there is not inherited and not permanent - it attaches to a job, mostly cardinals resident in Rome, diplomats of the Holy See and members of the Swiss Guard, and it ends when the appointment does. Nobody is born a citizen, and someone who loses it and holds no other nationality reverts to Italian citizenship, which the Lateran Treaty provides for so that nobody is made stateless.

It is the smallest country in the world at about 0.44 square kilometres, with a population of a few hundred, and it exists because of a sixty-year standoff. When Italy annexed the Papal States in 1870 the popes refused to accept it and stayed inside the Vatican calling themselves prisoners, which was only settled in 1929.

The oddities follow from the arithmetic. Per head it has the highest crime rate in the world, because a few hundred residents are divided into the pickpocketing of millions of visitors. It has no hospitals, so nobody is born there in the ordinary course of things, and no taxes, being funded by donations, investments, and the museums. The Swiss Guard are the only permanent element of what was once a much larger arrangement of papal armies, retained in 1970 when the rest were disbanded.