Is a person on a f1 or hb1 visa responsible for federal and fica tax withholding? If not, what is the responsibility of the employer for tracking the continued eligibility of the employee?

On F1/OPT you are exempt from residency for 5-years and you do not pay FICA taxes.
On H1-B, which is work visa, you must pay FICA taxes. It is employer's responsibility to deduct FICA taxes if you are an employee.
Read about F1 or H1-B tax filing: Your U.S. Tax Return: The U.S. Visas

Thank you for the information. It is very helpful. Is it the responsibility of the employer to track the eligibility of the employee?

It is the employer's responsibility to withhold the required taxes, and to ask all pertinent questions to ensure that the proper withholdings ae being done.

It is the employee's responsibility to ensure that the employer is aware of his visa status and knows which taxes should be withheld.

Thanks for your response.

Glad to help!