Yes, if you claim your son as a dependent, then the penalty for lack of health insurance and repayment of advance premium tax credits are yours.
The way the Affordable Care Act is set up, the health insurance requirement applies to everyone in the "tax household" - that is, everyone named in the tax return. If your son did not have insurance, and he is listed as your dependent, then you pay the penalty.
Likewise, the calculation for the advance premium tax credit depends upon your household's income if he is your dependent, and it would include his if he earned more than the filing threshold in 2016. However, if he earned more than $4050 and was over 19 and not a full-time student, then you couldn't claim him as your dependent.
According to the IRS, For purposes of Form 8965, your household income is your modified
adjusted gross income (MAGI) plus the MAGI of each
individual in your tax household whom you claim as a
dependent if that individual is required to file a tax return because
his or her income meets the income tax return
filing threshold. Use the Filing Requirements for Children and Other
Dependents
chart to determine whether your dependent is
required to file his or her own tax return.
If you don't list your son as your dependent, then you will lose his exemption of $4050 and possibly Head of Household status if he was your qualifying child. Only you can decide which outcome is the best for you.
Check the instructions for form 8965 for more information.
