Fortunately the UK website is quite comprehensive and useful in this case. If you wish to leave the airport i.e you exit out of the airport (in this case to travel to another airport), this is called a Landside transit. You will need a visa in order to do so i.e for the LTN to LHR transit.
As pointed out by @AndreyChernyakhovskiy, unfortunately the case of Ireland is a little bit more complicated than you would imagine. This is due to the CTA or Common Travel Area agreement. Underneath CTA, the flights leaving from British Airports to Ireland are considered short-haul international flights.
Thus, they depart from the domestic terminal instead. In order to do this you will need a landside transit visa. More information can be found on the Heathrow Airport website and the Gatwick Airport website which clearly state this information as well.
If you were an Iranian Citizen traveling to Ireland somehow without passing thru border control i.e doing an airside transit, this link would have applied to you.
One thing to note here is the exemptions. You would have been exempted from the visa if you were to have:
a valid uniform format category D visa for entry to a state in
the European Economic Area (EEA)
Ireland counts as a state in the EEA and thus if you have a category D visa for Ireland (i.e a long stay visa) , you don't not need a Direct Airside Transit Visa for the UK.
In all other cases, otherwise you will need a landside visa and as far as I can see, the only way to get to Ireland from the UK involves passing thru border control at the airport.