To a great degree this depends upon the country in which the person is taking refuge. In Canada, residents, whether under a labour permit, permanent residency, citizenship, or approved refugee status are eligible for universal free health care. recently, the Tory government has eliminated health benefits for people claiming refugee status whose status has yet to be approved.
In the United States, as health care is offered by a variety of different providers, much depends on whether the refugee has employer supplied health insurance. Refugees whose applications have been approved in the United States are eligible for the same benefits as permanent residents.