The Tur (Rabbenu Yaakov Ba’al Ha’turim, Germany-Spain, 1270-1340), in Orah Haim (Siman 585), cites his brother, Rabbenu Yehiel, as questioning the custom he observed among the Spanish Jewish communities to hire somebody to blow the Shofar on Rosh Hashanah (listen to audio recording for precise citation). It is forbidden to accept payment for services rendered on Shabbat or Yom Tob, and it would thus seem forbidden for somebody to receive payment for blowing the Shofar on Rosh Hashanah.
The Shulhan Aruch (Orah Haim 585:5) addresses this issue, and rules that if a person accepts payment to blow the Shofar on Rosh Hashanah, or to serve as Hazan or explain the Parasha on Shabbat or Yom Tob, he will not enjoy blessing from that money. Later authorities, including Hacham Ovadia Yosef (Yabia Omer, vol. 5, Orah Haim 25), note that the Shulhan Aruch does not forbid accepting payment for performing these services on Shabbat or Yom Tob, but merely warns that it is unadvisable. This ruling of the Shulhan Aruch thus appears to contradict an earlier passage (306:5) in which he rules that a Hazan may not be hired to lead the services on Shabbat, and then cites a view that allows such arrangements. There is a famous principle of "Setam Va’yesh Halacha Ki’stam," which means that when the Shulhan Aruch issues a ruling, and then cites a different view, he accepts as Halacha the first view. Seemingly, then, he sides with the position that forbids hiring people to serve in the synagogue on Shabbat. Yet, in the later passage, mentioned above, he allows doing so, albeit with a warning that the recipient of payment will not enjoy blessing from the money.
Hacham Ovadia Yosef writes that we should follow the "Mishna Aharona," meaning, the later of these two contradictory passages in the Shulhan Aruch, which allows hiring people to perform synagogue functions on Shabbat and Yom Tob. Indeed, as Hacham Ovadia observes in his work Yehave Da’at (vol. 1, Siman 53), this is the accepted custom. This is also the ruling of the Mishna Berura (commentary by Rav Yisrael Meir Kagan of Radin, 1839-1933).
Similarly, Rav Moshe Halevi (Israel, 1961-2001), in his work Menuhat Ahaba (vol. 1, 10:27), writes that it is permissible to accept payment to read the Torah or lead the services on Shabbat, or blow the Shofar on Rosh Hashanah, because the Sages did not forbid accepting payment for services involving a Misva. Nevertheless, he adds, in order to satisfy all opinions, there are those who make a point of performing some kind of service for the synagogue on a weekday, as well. For example, somebody who is hired to lead the services on Shabbat should, on at least one occasion, lead the services on a weekday. This way, the payment he receives will cover both work performed on Shabbat and work performed on a weekday, and Halacha permits accepting payment for a job that is done on both weekdays and Shabbat (a provision known in Halacha as "Habla’a").
As for the final Halacha, then, it is, strictly speaking, permissible for somebody to perform a service in the synagogue on Shabbat for pay. Preferably, however, he should perform a service during the week, as well. Thus, youth group leaders who are hired to run programs for the youth on Shabbat should, preferably, be asked to buy games or other materials for the group during the week. Likewise, a Ba’al Keri’a who is hired to read the Torah on Shabbat should be asked to read occasionally during the week.
Summary: It is permissible for a synagogue to hire somebody to perform a service for the congregation on Shabbat or Yom Tob, such as a Hazan, Ba’al Keri’a, Ba’al Teki’a, or youth group leaders. Preferably, they should also be given some responsibility to perform occasionally during the week, so that they are not paid solely for work done on Shabbat or Yom Tob.