This is part of the welfare state-based objection to immigration. The broad outline of the argument (in the US context) is as follows:

• Many immigrants, particularly illegal immigrants lack health insurance and lack the financial resources to pay for their own health care.
• As a result, when they need to avail of health care, they typically do so using emergency medical care provisions in hospitals. Emergency care units are bound by law and ethics not to turn away patients, regardless of their willingness or ability to pay.
• While the most direct brunt of the costs is borne by hospitals, the costs are passed on to the general population in two ways: (i) as taxpayers, they fund the goverment subsidies to the hospitals, (ii) the increased costs of operation may be passed on to insurance companies and hence to insurance purchasers in the form of higher premiums.

The exact framing of the argument would differ somewhat in countries with universal health care systems, but the broad concerns would be similar.

For more on the US-specific version, see US-specific emergency medical care for immigrants.

Responses by open borders advocates

• The Puzzling Ethics of Emergency Care where Bryan Caplan considers the conflicting moral intuitions that make people oppose the right to migrate but support tax-funded emergency medical care for immigrants, including illegal immigrants.
• The starving Marvin hypothetical, while it does not directly deal with emergency medical care, does deal with the paradoxical stance of forbidding somebody from immigrating because if that person did, you would feel obliged to help that person.