Kohenim serving in the Beit haMikdash were divided into mishmarot. Do we know which one the descendants of Eli Kohen Gadol and his sons were members of?
The following is only a partial answer. I am not aware of any explicit source that states what mishmar Eli and his descendants were part of. According to the gemara in Taanit 27a and the Tosefta Taanit 3:2, there were originally eight mishmarot organized by Moshe, four from Elazar and four from Itamar. According to the gemara, Shmuel doubled them to sixteen, and then David made them into twenty-four. Not much is known about the pre-Davidic mishmarot, so I don't think there's any way to know which mishmar Eli himself was a part of, but evidently he was part of one of the four Itamaric mishmarot, as Eli was a descendent of Itamar (according to Divrei Hayamim 1:24:3, Achimelech, a descendant of Eli, was also a descendant of Itamar). His descendants until the creation of the sixteen mishmarot would have belonged to the same mishmar as he did, and later descendants, until the creation of the twenty-four, would have belonged to one of the new Itamaric mishmarot, though it is unclear how many there were (was the number split evenly between Itamar and Elazar or was it already a case of a 2:1 ratio?).
Now, about his descendants from the time of the twenty-four: We can minimize the options to the eight mishmarot who were descendants of Itamar. According to the Yerushalmi in Taanit 20a, the second set of eight mishmarot were from Itamar, which gives us (Divrei Hayamim 1:24:11-14):
ישוע, שכניהו, אלישיב, יקים, חפה, ישבאב, בלגה, אימר
Yeshua, Sh'chanyahu, Elyashiv, Yakim, Chupah, Yeshve'av, Bilgah, Imer
Hypothetically, it might be possible to narrow down the list more, if there was a way to connect one of the descendants of Eli mentioned in Talmudic sources to one of the mishmarot somehow. After the destruction of the Temple, the mishmarot escaped to the Galilee and settled in different towns (see here), so maybe, just maybe, if one of the descendants was found to be connected to one of these towns, then perhaps that would signify that he was a member of that mishmar. These are the descendants I'm aware of: Rabbah (bar Nachmani), Abayei, Khaylil (Abayei's father and Rabbah's brother), Rav Bivai bar Abayei, "the family of Rabban Yochanan ben Zakkai" (Rosh Hashanah 18a), and (some of?) the population of the town of Mamlah (Beresheet Rabbah 59:1).
I found two possibilities for connecting some of the descendants to kohen towns in the Galilee:
In Shaar Hagilgulim, Introduction 38, it is said that the graves of Abayei and Rava are to be found in "Avenit", which is identified with Yevanit, the location of Mishmar Imer. Coincidentally (or not), Imer was one of the mishmarot descended from Itamar.
According to various sources compiled by Rabbi Shmuel Klein (see link about the mishmarot escaping to the Galilee), the population of Mamlah or Mamliach were descendants of Eli. Elsewhere, it says that Chezir settled there. Chezir wasn't a descendant of Itamar. Perhaps the two groups shared the town, though how they came to be connected is unclear (also noted in this essay, pg. 82, note 40). According to Rashbam on Bava Batra 137b, the combination of the town's name and the defects of the descendants of Eli were used to criticize Rav Bivai bar Abayei.1
In short, there seems to be a bit of case for maybe connecting Bnei Eli to the mishmar of Imer, but this is based on a lot of conjecture. A bit of a stronger case may be made to connect them to Chezir, but at the same time, it seems that Chezir was a descendant of Elazar, so the connection remains mysterious.
An entirely different option was suggested by Rabbi Avraham Taub in his commentary on Shmuel, "Divrei Tova" (דברי טובה). On the verse:
"וְהָיָה כָּל הַנּוֹתָר בְּבֵיתְךָ יָבוֹא לְהִשְׁתַּחֲוֹת לוֹ לַאֲגוֹרַת כֶּסֶף וְכִכַּר לָחֶם וְאָמַר סְפָחֵנִי נָא אֶל אַחַת הַכְּהֻנּוֹת לֶאֱכֹל פַּת לָחֶם."
"And all the survivors of your house shall come and bow low to him for the sake of a money fee and a loaf of bread, and shall say, ‘Please, assign me to one of the priestly duties, that I may have a morsel of bread to eat.’”" (Shmuel 1:2:36)
he writes (my translation):
"It is possible that when David, along with Tzadok the kohen and Achimelech son of Evyatar organized the twenty-four mishmarot kehunah...they did not assign from the sons of Eli any mishmar of kehunah, because the sons of Eli could not be trusted in the service of the kehunah due to the fact that they died young, and we find in Divrei Hayamim 1:9:22 "All these, who were selected as gatekeepers at the thresholds, were 212. They were selected by genealogies in their villages. David and Samuel the seer established them in their office of trust" - it seems that the basis of the assignments was the trust that this person would serve dutifully, but the sons of Eli could not trust the dutiful continuation of their service, and so there was not a mishmeret of service from the sons of Eli...and Rashi wrote that "memulai" - a miserable family, from the House of Eli and the gemara there says this about Rav Bivai bar Abayei that since he came from a memulai family, he said things that were wrong, for the descendants of Eli were a miserable family because they didn't have a tradition for Torah and wisdom because they died young, and for this reason they could not be tasked with forming a mishmeret of kohanim because they could not properly pass on from one another the tradition of the priestly service..."
In short, due to their curse, the sons of Eli were never assigned to a mishmar. This makes some sense in light of views that the kohanim of Anatot were descendants of Evyatar who was removed from service (see here and here for example), but on the other hand, none other than his son Achimelech is in charge of setting up the mishmarot - it seems strange that he would not include his own family (i.e., it makes more sense to associate the removal of some of Bnei Eli only to when Shlomo removed Evyatar from service, and not the removal of all of Bnei Eli, already in the time of David).
1 I came across now an essay by Prof. Aharon Demsky ('Abbaye’s Family Origins: A Study in Rabbinic Genealogy', in: "Follow the Wise": Studies in Jewish History and Culture in Honor of Lee I. Levine, pp. 235-240) where he suggested the same conjecture regarding Abayei's family origins and noted the difficulty with Chezir being a descendant of Elazar according to the Yerushalmi but raised the possibility of simply disagreeing with the Yerushalmi's drasha about the name Chezir (="returned", i.e., returned to the Elazar families). There's some logic for his suggestion, as it would put all the Elazar families in one group and the Itamar families wouldn't be pushed in-between. Per his suggestion, Chezir would be the first of the Itamar families (being no. 17 in the list of mishmarot).