I generally assume that everyone knows how to use Wikipedia, but it seems not. From today’s Age:
Reports suggest that in the 19 American presidential elections between 1888 and 1960, the candidate with a height advantage won all but one.
As Wikipedia notes, between 1888 and 1960, Coolidge, McKinley (twice) and Harrison all beat their taller opponents.
For the 47 elections in which the heights of both candidates are known, the taller candidate won 29 times (approximately 62% of the time), the shorter candidate won 15 times (approximately 32% of the time), and the candidates were the same height three times (about 6% of the time). If the scope of elections is restricted to 1900 and on, only 6 out of 28 elections were won by the shorter candidate.
Update: Regular reader Alistair Campbell takes up the challenge, posting a Google spreadsheet with heights for the PM and Opposition leader. Anyone with more information, please update it (I suspect there is much that is hidden in biographies).