To the UK:
If you are transferring your household goods to the UK related to a change in residence, you may be liable for customs charges unless you apply for transfer of residence relief from HMRC in advance.
The UK guidance on this can be found here.
From the UK to the EU:
If you are moving your household from outside the EU, e.g. the UK, to an EU country such as Ireland, you may be liable for VAT and/or customs charges on the carriage of your household goods. We recommend that you consult the Irish revenue in relation to transfer of residence from outside the EU here for advice and guidance.