Melanie,
I think the Templepatrick in question is in Co. Antrim. There are trees on Ancestry which show Robert Greer dying in Cloghanduff, a townland near Templepatrick, on 25.9.1821. (Though no supporting evidence provided). Cloghanduff today is just to the east of Templepatrick village. The modern Cloghanduff road runs through part of it. Mainly agricultural land with some residential housing. In 1901 Cloghanduff had 31 homes and a population of 104. Mostly engaged in farming or agricultural work. Almost all were Presbyterian which reflects the population make up in that part of Ireland. So they’ll mostly be descendants of Scots who settled in the area in the 1600s.
No Greers in the townland in Griffiths Valuation c 1860 and none in the parish in the tithes c 1833.
https://www.irishgenealogyhub.com/antrim/tithe-applotments/templepatrick-parish.php
Belfast Newsletter of 27th September 1768 reports Arthur Greer having 17 acres, 1 rood and 32 perches of land (Irish plantation measure) in Cloghanduff. His landlord was the Castle Upton estate and they were advertising for sale the superior leases of a whole batch of farms across quite a few townlands in the Templepatrick area. The individual tenants would have been unaffected. It just meant they would have acquired a new landlord.
It looks as though the Greer family were gone by the time of the tithe applotment records. Presumably the farm was sold after Robert’s death in 1821.
The family may have attended what is now Templepatrick Non Subscribing Presbyterian church, though prior to 1830 it was just the Presbyterian church. It was established in the 1640s and is one of the oldest Presbyterian congregations in Ireland. It has records starting in 1796. There’s a copy of those records in PRONI (the public record office) in Belfast. PRONI’s catalogue says they have:
[Prior to 1830, when a branch of the Church split away to form the Non-Subscribing Presbyterian Church, the records relate to Templepatrick Presbyterian Church of the General Synod] Baptisms, 1796-1980; marriages, 1797- 1848; burials, 1966-78; lists of communicants, 1811-36; accounts, 1799-1812; list of persons in each family connected with the church, 1854-5; session minutes, 1646-1743 and 1878- 1962.
The following graveyards are in the parish: Craigarogan, Templepatrick Old, Templepatrick Church of Ireland, Mallusk & Umgall. Many Presbyterians were buried in Church of Ireland graveyards, especially if their own church didn’t have a graveyard. Presbyterians generally didn’t keep burial records, so you’ll have to hope there’s a legible gravestone. Since the family were evidently farmers, they should have been able to afford a gravestone.
PRONI has a record of an Arthur Greer dying around 1856. He lived in Templepatrick and probate was granted that year. Unfortunately the will itself was destroyed in the 1922 fire in Dublin, during the civil war, and no other information survives.