According to The Concise New Patridge Dictionary of Slang and Unconventional English the humorous expression dates back to 1985:

• more tea, vicar? used humourously to acknowledge a fart or a belch, UK 1985.

(probably from a sitcom popular at that time)

According to The Concise New Patridge Dictionary of Slang and Unconventional English the humorous expression dates back to 1985:

• More tea, vicar? used humourously to acknowledge a fart or a belch, UK 1985.

From The virtual linguist:

• I noticed in the kitchen department of John Lewis china teapots with the phrase "More tea, vicar?" written on them. I wondered if the phrase meant anything to the hundreds of tourists around. I doubt it. "More tea, vicar?" is a phrase associated with old sitcoms where vicars were always either bumbling old dodderers or naive simpletons. Something would happen in the storyline eg someone would start swearing, belch or be generally uncouth, at which point the genteel lady of the house would pick up the teapot and ask "More tea, vicar?" in a desperate attempt to divert his attention from the insalubrious goings-on. Outside of sitcoms someone who commits a faux pas, or gets himself/herself in a tricky or embarrassing situation while with a group of friends, might utter the phrase "More tea, vicar?" as a way of changing the subject. It is meant to be amusing, of course.

(virtuallinguist.typepad.com)