Jakub Mensik beat Shelton 5-7 6-4 6-3 to level the best-of-five series at 2-2 before Jiri Lehecka sent a roaring Prague crowd into raptures by beating Tien 6-3, 7-5 in the deciding singles.
It was a second straight loss for Shelton, a day after he was beaten by Lehecka.
American pair Christian Harrison and Austin Krajicek had made it 2-1 after winning their doubles match over Adam Pavlasek and Patrik Riki 5-7 6-3 7-6 (8-6).
Austria made the Final 8 after Jurji Rodionov beat Belgium's Zizou Bergs 7-6 6-3 to build an insurmountable 3-1 advantage. Austria also won the doubles match on the outdoor clay in Vienna, a day after they split the opening single matches with Belgium.
Toby Samuel and Wimbledon semi-finalist Arthur Fery won their singles matches to put Britain 2-0 up at home to Ecuador.
Alexander Zverev gave Germany a 1-0 lead over Croatia by beating Matej Dodig 6-4 7-6 (7-3) before Dino Prizmic leveled the series with a 6-4 6-3 win against Daniel Altmaier.
It was Zverev's first match since winning the US Open final last weekend against Shelton. That came after the 29-year-old Zverev won his first grand slam title at the French Open in June.
The second-ranked Zverev, who had 15 aces, dominated on the indoor hardcourt in Halle, Germany, with Dodig having no break-point chances.
Doubles and reverse singles are scheduled for Sunday in the best-of-five match series. The winning team advances to the Final 8 in Bologna, Italy, in November.
South Korea earlier became the first Asian nation to reach the Final 8 after clinching their tie against India 3-1.
The South Koreans came from a set down in both singles matches on Friday to lead 2-0.
They lost the opening doubles Saturday but Soonwoo Kwan beat Sumit Nagal 6-1 6-2 in the first reverse singles.
The Final 8 format was introduced in 2022.
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