Regulators will shut down Health Republic Insurance of New York, the largest of the nonprofit cooperatives created under the Affordable Care Act, in the latest sign of the financial pressures facing many insurers that participated in the law’s new marketplaces.

The insurer lost about $52.7 million in the first six months of this year, on top of a $77.5 million loss in 2014, according to regulatory filings. The move to wind down its operations was made jointly by officials from the federal Centers for Medicare & Medicaid Services;...