This is likely to be the Plume of Feathers in 1861. Listed at High Street in the 1901 census.
Burnham pub history index
Directory of Pubs in the UK, historical public houses, Taverns, Inns, Beer Houses and Hotels in Buckinghamshire. The Buckinghamshire listing uses information from census, Trade Directories and History to add licensees, bar staff, Lodgers and Visitors.
The following entries are in this format:
Year/Publican or other Resident/Relationship to Head and or Occupation/Age/Where Born/Source.
1861/Richard Barnett/Cordwainer/56/Swinbrook, Oxfordshire/Census
1861/Henrietta Barnett/Wife/54/Woburn, Buckinghamshire/Census
1861/Charles Barnett/Son/17/Burnham, Buckinghamshire/Census
1861/George Barnett/Son/13/Burnham, Buckinghamshire/Census
1861/Caroline/Daughter/11/Burnham, Buckinghamshire/Census
1861/Joseph Gregory/Lodger, Ag Labourer, Widow/69/Ardington,
Buckinghamshire/Census
1861/James Williams/Lodger, Ag Labourer/21/Marlboro, Wiltshire/Census
1861/Sarah Williams/Lodger Wife/21/Ogbourne, Wiltshire/Census
1871/Henrietta Barnett/Beerhouse Keeper/63/Woburn, Buckinghamshire/Census
1871/Charles Barnett/Son, Garden Labourer/27/Burnham, Buckinghamshire/Census
1887/Henry Siffleet/Beer Retailer/../../Kellys Directory
1891/Henry Siffleet/Beer Retailer/../../Kellys Directory
1891/Henry Siffleet/Beer Retailer/32/Newick, Sussex/Census
1891/Charlotte Siffleet/Wife/38/Tilehurst, Berkshire/Census
1891/Enid Siffleet/Daughter/5/Burnham, Buckinghamshire/Census
1891/Amy Siffleet/Daughter/4/Burnham, Buckinghamshire/Census
1891/Harry Siffleet/Son/3/Burnham, Buckinghamshire/Census
1895/Henry Siffleet/Beer Retailer/../../Kellys Directory
1901/Henry Siffleet/Beer Retailer/42/Newick, Surrey/Census
1901/Charlotte/Wife/48/Tilehurst, Buckinghamshire/Census
1901/Amy/Daughter/14/Burnham, Buckinghamshire/Census
1901/Ethel/Daughter/9/Burnham, Buckinghamshire/Census
1901/Thomas West/Boarder, Iron Finisher (Foundry), Widow/52/Reading,
Berkshire/Census
*** Provided By Colin Ager