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, 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).