Britain's Murray and Brazil's Soares were beaten 6-7 (4-7) 7-5 10-8 in Washington.
In the men's singles, Alexander Zverev overcame Kei Nishikori 6-3 6-4 to set up a final against Kevin Anderson, who beat Jack Sock by the same score.
Ekaterina Makarova will play Julia Goerges in the women's final.
Russian Makarova, who is coached by Briton Nigel Sears, won 3-6 6-3 6-4 against France's Oceane Dodin, while Goerges emerged with a 5-7 6-4 7-5 victory over Andrea Petkovic in their all-German tie.
Murray and Soares had saved seven match points in Friday's quarter-final win over James Cerretani and Marc Polmans.
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But Australia's Peers - Murray's former doubles partner - and Finland's Kontinen ended their bid despite losing the opening set.
Kontinen is second in the ATP men's doubles rankings, with Peers third, 31-year-old Murray fifth and Soares sixth.
Wimbledon champion Garbine Muguruza was knocked out of the Stanford Bank of the West Classic in California following a 6-3 6-2 semi-final defeat by Madison Keys.
American Keys claimed the first set by winning 12 of the last 14 points and went on to secure victory in less than an hour.
She will play compatriot Coco Vandeweghe in the final after she beat Catherine Bellis 6-3 6-1.
Vandeweghe, who lost to Serena Williams in the 2012 final at the event, hit seven aces and 30 winners in the match.
Jamie Murray and Bruno Soares lost their Citi Open men's doubles semi-final to top seeds John Peers and Henri Kontinen on a third-set tie-break.