We find in the Torah a prohibition against directly counting Jews and a warning that doing so can lead to a plague, Heaven forbid. In several instances in Tanach, Jews were counted indirectly to avoid this prohibition. For example, we are told in the Book of Shemuel that when King Shaul had to count his soldiers before going out to war, he had each soldier give a piece of pottery, and those pieces were counted to determine the number of troops.
Rabbi Eliezer Waldenberg (Jerusalem, 1915-2006) wrote a treatise that appears in his work Tzitz Eliezer (vol. 7), discussing the permissibility of the Israeli census. In this treatise, he establishes several rules governing the procedure for counting Jews. He writes that counting actual bodies is forbidden under all circumstances, even for the purpose of a Mitzva, such as counting men for a Minyan or in preparation for a Mitzva. When Jews must be counted for the purpose of a Mitzva, they should be counted indirectly. For this reason, the practice has evolved that when men are counted in the synagogue for a Minyan, rather than using numbers, we recite verses or phrases containing ten words, such as the verse "Hoshi'a Et Amecha…" Even when counting in this fashion, one should ensure not to point and count the people, and should rather just look at them and recite the words of the verse.
Rabbi Waldenberg adds that in a situation where Jews were counted in a forbidden fashion, the people involved should donate money to charity as atonement. They should donate at least the sum of a "Machatzit Ha'shekel" – which amounts to about $3-4 – the amount that Benei Yisrael would give as a means of counting the population.
Rabbi Moshe Feinstein (Russia-New York, 1895-1986), in his work Iggerot Moshe (Yoreh Dei'a, 117:2), addresses the question of whether a cemetery may place numbers on the tombstones to allow visitors to find their loved ones' graves more easily. Placing numbers on the tombstones amounts to counting the deceased people buried in the cemetery, and Halacha forbids acting towards the dead in a manner that is forbidden for live people, as this can be seen as taunting them for their inability to perform Mitzvot. (This Halacha is known as "Lo'eig La'rash" – "ridiculing the impoverished," which refers to a deceased person, who is "impoverished" in the sense of Mitzva observance.) Therefore, Rabbi Feinstein indeed rules that a cemetery may not place numbers on tombstones.
Summary: One may not conduct a headcount of Jews under any circumstances. For purposes of a Mitzva, one may count Jews indirectly, without actually counting bodies. Cemeteries may not place numbers on tombstones, as this amounts to "counting" Jews, as well.
