Whether PRSI applies to deposit interest will depend on whether you are an employee or self employed. If you are self employed (with a sufficient level of income) PRSI applies and will be due as part of your income tax payment under self assessment for the year.
If you are a PAYE employee you are whats called an "excepted self employed contributor" and will not be liable to PRSI on unearned income including deposit interest.
This exception for employees does not apply to the Health Levy. However, my understanding is that unless a tax return is filed by the PAYE employee due to level of non-PAYE income, Revenue do not seek to recover the health levy amount.