I am travelling to Europe using an Irish passport and a UK EHIC. Is there a problem with this, as
I receive healthcare from the NHS. Living in N Ireland I can claim dual citizenship, but I wonder if I
should obtain a UK passport.
I doubt if there is any grounds for worry - you are presumably living in the Uk and therefore entitled to healthcare there, your nationality has little or no relevance particularly as you are a citizen of an EU member country. Nationals of EU countries can generally live anywhere within the EU and contribute to schemes, etc. anywhere in the EU.