The House Republican effort to overhaul the Affordable Care Act could hit many rural areas particularly hard, according to a new analysis, sharply increasing the cost for some residents buying their own insurance.

In extreme cases, the amount a consumer might owe for a plan could exceed that person’s annual income. In Nebraska’s Chase County, a 62-year-old currently earning about $18,000 a year could pay nearly $20,000 annually to get health-insurance coverage under the House GOP plan—compared with about $760 a year that person...