I'm going to assume that you will be picking up and returning the rental to the same location in Ireland.
Some companies may allow cars rented in Ireland to travel to the United Kingdom (for an additional fee); others do not allow it.
For example, I checked driving restrictions by checking a mock reservation/rate quote for Dublin pickup and return, compact car, with Avis, Budget and Hertz: (company links open in a new window)According to Avis.com
Cars rented in Ireland may not be driven into to the U.K. or continental Europe. They are only allowed to travel into Northern Ireland.
NO travel to the UK permitted. Vehicles rented in Ireland may be taken into Northern Ireland only.
A vehicle rented in Ireland can be driven into Northern Ireland with no restrictions.
The rented car is allowed to be driven to the UK, but not to any other country.
If a renter plans on driving into the U.K., they must tell the Hertz counter at the time of rental and are required to buy an AA 5-star cover at any AA office prior to leaving Ireland.
Make sure to check on Congestion Charging if you intend to drive in Central London.
You should check the terms and conditions... specifically driving restrictions, for any rental car company that you are considering. A link to view terms and conditions is generally available during the rate quote process on a car rental company website. Or, you can contact a reservation rep via a rental company's toll free number to clarify policies for a specific rental location.
Hope this helps!