Nigel Rees actually published a book titled More Tea Vicar? An Embarrassment of Domestic Catchphrases (2009), but its entry for that phrase is disappointingly vague:
more tea, Vicar? A correspondent who, understandably, wished to remain anonymous advanced the family phrase, 'for after a fart, or to cover any kind of embarrassment'. Paul Beale has collected various forms for a revision of Partridge/Catch Phrases, including 'good evening, vicar!'; 'no swearing please, vicar' (said facetiously to introduce a note of the mock highbrow into a conversation full of expletives); 'another cucumber sandwich, vicar' (after an involuntary belch); 'speak up, Padre!/Brown/Ginger (you're through)' (as a response to a fart).
Rees (again) in A Word in Your Shell-like (2004) provides additional information on the phrase:
more tea, Vicar? Phrase for use after a fart or to cover any kind of embarrassment. British use, from the 1920s/30s? ... David Rogers declares: 'The phrase, "More tea, Vicar?" has entered the language as shorthand for comfortable suburbia.' Hence these stories: '"More tea, Vicar?" asked Lady Lavinia as she poured the tea with her other hand' and 'One day the young Vicar was visiting two elderly ladies. Whilst he was sitting on the shiny sofa, he passed wind mightily and noisily. As the echoes died away, one of the ladies filled the embarrassing silence by asking, "More tea, Vicar?" "Oh no!" he replied," "It makes me fart!"'
On the other hand, Julia Cresswell, The Cat's Pyjamas: The Penguin Book of Clichés (2000) indicates that "More tea, vicar?" was a catchphrase associated with sedate gentility before it got commandeered by jokesters:
The tea party expression [which arises in connection with "behaviour that would make something less outrageous look like a vicarage tea party"], and the associated catchphrase More tea, vicar?, have been in use as a comparative standard of innocence since at least the 1950s.
In any case, a Google Books search doesn't turn up any matches at all for "more tea, vicar" before 1981, when the phrase appears in New Zealand Alpine Journal, volume 34, not in the context of farting or complacent gentility but of a surprisingly easy ascent:
Well, the photos we took from the air must have been tilted, because we wandered up in perfect conditions as though it was a Sunday picnic. More tea, Vicar!
Under the circumstances, I think that the circa 1985 date put forward by The Concise New Partridge Dictionary of Slang and Unconventional English (2015), noted in Josh61's answer, is far closer to the mark than Nigel Rees's "from the 1920s/30s?" as the starting point for saying "More tea, vicar?" after someone farts.