WASHINGTON—Industry groups are ramping up a campaign to repeal or delay two Affordable Care Act taxes that have critics in both parties, the latest effort to tackle the health-care system following the collapse of the broad Republican push to repeal the Obama-era law.

The taxes, one on medical devices and another on health-insurance plans, were previously delayed in late 2015 as part of a larger spending package. They now are set to take effect beginning in 2018, unless Congress postpones them again or kills them off entirely.

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