Originally Posted by
dulciusexasperis
Regarding a US visa waiver, what has a US visa waiver got to do with what I wrote? Some countries require that you pay for a visa or for a visa waiver, like the USA. Some countries make you pay for a visa but not for a visa waiver. Therefore, there are cases where entering on a passport that gets you a free visa waiver entry is advantageous. Do you not understand that or do you think talking about a country which requires you to pay for a visa waiver means there is never an instance where you can save a few dollars entering a 'free' visa waiver country? In regards to this specific post, there is no US Visa Waiver involved as the traveller is a US Citizen.
This isn't reflected in your original post which suggested that visa waivers, in general, "cost nothing".
Moreover, I can't make too much sense of your "passport may be flagged as having overstayed". The UK (currently) couldn't reject the OP as Irish citizens are, save from a few extremely narrow exceptions, free to enter and leave the UK as they please. The US, on the other hand, might remind the OP to use their US passport but can't really do much more. They certainly can't accuse a US citizen of overstaying in the US. The risk of overstaying subsequently only applies to third countries, which gets us way off-topic from the initial query.
As far as travelling with dual passports between the two countries issuing the passports is concerned, the general advice is to use the passports of each country to enter and leave the issuing country. Presenting both passports at check-in will make sure that the airline can check that you're okay to travel. Obviously, that method will also enable the OP to use EEA lines, including e-gates, upon arrival in the UK.