Moving furniture and household items from the UK to Ireland (or the other way) can be costly.
For smaller moves - single people or couples - you could try using a "Man and Van "type service.
Prices for a full load on a large Transit type van - with a driver can range from €600 to €900 . If the driver has a load to pick up on the other side - it will reduce the price . So it helps if you are flexible on dates so the driver can arrange trips when there is a load going both ways .
You could hire a van and do it yourself : - Van hire (Ford Transit 350) with insurance etc can cost around €50 a day . You will probably need to hire it for 3 days minimum and when you add in ferry costs and fuel you won't get much change out of €600 .
You can hire Vans in the UK from Europcar or Budget or Hertz
Large removals will require the services of a proper removal company - with prices depending on the size of your load. Expect to pay in the region of £1200 to £1500 or more for an average sized house.
Prices are cheaper if you do your own packing and dismantling. Make sure the company provides insurance.