Just to clarify an earlier posting,
The purpose of an EHIC is to cover you for 'necessary' healthcare while temporarily in another EU state - I never said that it would cover anyone to travel abroad for the purpose of obtaining medical care. These are very different matters.
The EHIC card only provides care in the public health system in another EU state and is mainly used to provide necessary care if someone becomes suddenly ill, is injured in an accident or requires other 'necessary' care while they are abroad temporarily. For example, on holiday, a weekend break, etc.
It is of course possible to be sent abroad for medical treatment. If you are attending a family doctor/consultant in a private capacity and are referred abroad for treatment then you (or your insurance company) will have to pay the cost of this treatment.
If you are attending a family doctor/consultant as a public patient it is possible to be referred abroad for treatment but usually this is only where the doctor signs a form (i.e. clinical decision) and this treatment is not available in Ireland through the public health service. The form is then returned to the HSE Area who will make a decision on the case and then arrange treatment (or not).
Given the OP is considering travelling to Hungary for dental work, you should be aware that the EHIC will only cover emergency dental treatment, free of charge. Travelling over to Hungary for the purpose of obtaining dental treatment is not the same thing and I don't believe your EHIC would cover you for this. The reason I advocate anyone travelling in the EU to obtain an EHIC is you never know what could happen when you are away so it is wise to hold an EHIC anyway. I also always advocate that people hold travel insurance when travelling abroad. As useful as an EHIC is in terms of obtaining healthcare abroad - it will not meet the costs of repatriation if you need to be sent home - a useful pointer if you are seriously injured abroad.
In terms of paying therefore for this dental treatment, I think the options are checking whether it is covered by your private health insurance policy, or paying the cost abroad and claiming here under the Treatment Benefit Scheme. Again, you should check that you qualify for the Treatment Benefit Scheme in advance of travel if you intend to do this.