Judi,
I have not been able to confirm where James Whittaker (c1795 - 1873, Lanesboro) is buried. He could be buried at Lanesboro Church of Ireland Cemetery, without a tombstone.
However, I found the following which may be of interest:
Married at St. Michan's Church, Dublin City on 9th May 1870 by Rev. Marcus J. Bickerstaff (family connected to County Longford)
John Munn Whittaker, full age, bachelor, Lance Corporal 56th Regiment, 14 Linenhall Street, son of Edmond Whittaker (Tailor)
Mary Anne Pearson, full age, spinster, -, same place, daughter of Patrick Pearson, Bricklayer
witnesses: William Geraghty and Elizabeth Plunkett
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_re…
Death Certificate, 25th November 1873, Lanesboro, Edward Whittaker, Male, Married, aged 70, Tailor, Natural Causes, Information received from Nicholas McGaver, Coroner for the County Longford, Registered 2nd December 1873 by J. Slevin, Registrar
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_retu…
Baptised sub-conditione 13th October 1876, New Ross RC Parish, County Wexford by Rev. D. J. Gorman
James Whittaker (born 16 June 1849), Ross
son of Edmund Whittaker & Margaret McGuinness
sponsor: Moses Duffy
[James Whittaker was an adult convert to Catholicism, in advance of his marriage below]
[There were a number of McGuinness families residing in County Longford at the time of the 1854 Griffith's Primary Valuation]
Married 4th September 1877, New Ross Chapel, County Wexford by Rev. P. M. Furlong C.C.
James Whittaker, full age, bachelor, Police, Taghmon, Edmond Whittaker, Pensioner
Ellen Teresa Byrne, full age, spinster, -, New Ross, John Byrne, Carpenter
witnesses: Denis Falvey and Mary Murphy
Registered 17th September 1877 by John Barron, Deputy Registrar
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_re…
1901 Census - 4 Tram Cottages, New Kilmainham Electoral Division, South Dublin, County Dublin
James Whittaker, Head of Family, Catholic, Aged 52, foreman labourer, married, born in Lancashire, England
Ellen Teresa Whittaker, Wife, Aged 42, -, married, born in County Wexford
children: Edmond (18), Margaret (16), James (15), Leo (13), Frederick (12), Arthur (8), Josephine (6) - all born in County Kilkenny
http://www.census.nationalarchives.ie/reels/nai003702904/
1911 Census - 4 Tramway Cottages, New Kilmainham Electoral Division, South Dublin, County Dublin
James Whittaker, Head of Family, Catholic, Aged 62, timekeeper Dublin United Tramway, married, born in Lancashire, England
Ellen Teresa Whittaker, Wife, Aged 57, -, married, born in County Wexford
(married 33 years, had 12 children born alive, 8 of whom were alive in 1911)
children:
Edmond (28), Margaret (26), Frederick (22) - born in County Kilkenny
Arthur (18), Josephine (16) - born in County Waterford
http://www.census.nationalarchives.ie/reels/nai000132042/
Death Certificate 6th May 1916, 4 Tram Terrace, Inchicore, Kilmaiham, Dublin
James Whittaker, male, married, aged 67, timekeeper, Bronchitis - one month certified, Jsephine Whittaker - present at death - 4 Tram Terrach, Inchicore
Registered 11th May 1916 by R. M. Power, Registrar
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_retu…
Royal Irish Constabulary Pensions:
James Whittaker, Constable when pensioned, County Kilkenny, aged 44 when pensioned, annual pension £42-2s-4d
Date of commencement of pension 16th September 1892, paid at the Pay Office, Died 6th May 1916, Paid to Ellen T. Whittaker, Widow
Hoping some of the above information is new to you,