Kohlschreiber, who had been going for an unprecedented fourth title at the clay-court tournament, failed to convert any of his three break points and lost to Horacio Zeballos of Argentina 7-6 (5), 6-4 in the second round.
"I can't be too hard on myself. He did well," Kohlschreiber said.
Monfils, who received a first-round bye, lost to Chung Hyeon of South Korea 6-4, 6-2.
The 78th-ranked Chung will next play Martin Klizan of Slovakia, who upset sixth-seeded Mischa Zverev of Germany 6-2, 6-1 in 49 minutes.
"To be honest, I didn't have a chance against him," said the older Zverev brother. Alexander "Sascha" Zverev is through to the quarterfinals.
Also, fourth-seeded Fabio Fognini lost to Argentine qualifier Guido Pella 6-3, 6-2. Pella will meet Zeballos in the quarterfinals.