Defending champion Murray was scheduled to play Italy's Fabio Fognini in the last 16, while 2008 winner Rafael Nadal was set to face France's Gilles Simon.
British number one Konta saw her quarter-final against Germany's Angelique Kerber postponed.
Konta and Jamie Murray had been due to start their mixed doubles campaign.
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Andy Murray, Johanna Konta and Rafael Nadal were among those frustrated on Wednesday as rain prevented play at the Olympic Tennis Centre.