I have experience with this.
For PRSI cover you need to have been working a set amount of time, so a 16 year-old would not have had enough contributions for cover. The HSE, despite what they say, pretty much wash their hands of children needing non-emergency treatment. The dental hospital is an option, but while the treatment is heavily subsidised you do pay for it with your time. Depending on the stage of the student treating you, you could be in their for 2 hours for a filling that a dentist would take 10 minutes for. Unfortunately for most people there is no option other than paying privately.
It is disgraceful how young people and full time students are expected to pay full private fees for basic dental care in this country.