There is currently a tax border between Ireland and Northern Ireland. 

The tax border between Ireland and Northern Ireland is an excise border.

There is no *customs* border between Ireland and Northern Ireland while the UK remains in the EU because membership of the European Union involves membership of a customs union: the European Union Customs Union with its Common External Tariff.

Excise is the application of a duty (colloquially "a tax") usually levied at the point of production on goods that are widely available. Excise rates are published [online][1]. 

Goods that commonly incur excise duty are tobacco, fuel and alcohol.

To avoid border movements whose purpose is to illegally circumvent duty an excise border is necessary between Ireland and Northern Ireland.

On the island of Ireland this tax border is managed without physical infrastructure for reasons of political tension. "Hardening" the border to mitigate fraud on the Ireland/Northern Ireland tax border is not possible.

**For legal movements of goods** duty revenue is collected via online payment system. For the UK this is the [Excise Payment Security System][2] (EPSS).

However, duty goods [do not incur][3] additional duty charges when crossing the Irish border if:

 - you transport them yourself
 - will use them yourself or give them away as a gift
 - have paid duty and tax in the country where you bought them

Another answer here highlighted the loss of revenue to the UK exchequer of up to [£200 million][4] each year due to fuel meeting these criteria crossing the border. Despite the headline it is *perfectly legal* for residents on one side of the border to fill up their vehicles on the other. The respective exchequers have no legal right to demand additional duty.

The focus for the tax border between Ireland and Northern Ireland is therefore the movement of duty goods for *resale*, usually in larger volumes. 

**For illegal movements of goods** the tax border between Ireland and Northern Ireland is managed through North/South police service cooperation: the PSNI in the North and the Garda Síochána in the South.

Intelligence is shared and [coordinated cross-border operations][5] (document found by @origimbo) are conducted, including but not limited to:

 - interdiction of illegal shipments/transhipments of duty goods
 - targeting of persons involved in such operations
 - use of marker systems to assist detection (eg. [Accutrace S10][6] for fuel)

An important argument for having a customs partnership "backstop" if the UK leaves the European Union is that a tax border (in that case for customs) on the island of Ireland would necessitate a hard border and exacerbate political tensions.

There is currently a tax border on the island.

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**Summary of when excise duty (tax) is due at the border**

If you buy a barrel of petrol & move it yourself over the border for personal consumption or for gifting then no additional duty is payable. 

If you buy a barrel of petrol & have it moved for you over the border - for whatever reason - then duty is payable via EPSS. 

If you buy a barrel of petrol for resale & move it over the border then duty is payable when the fuel exits a duty-suspension regime. eg. it is resold. Presumably there is a way to declare the taxable import.

  [1]: https://www.gov.uk/trade-tariff
  [2]: https://www.gov.uk/guidance/pay-excise-duty-without-a-guarantee
  [3]: https://www.gov.uk/duty-free-goods/arrivals-from-eu-countries
  [4]: https://www.belfasttelegraph.co.uk/news/northern-ireland/buying-fuel-in-republic-costs-uk-government-200m-35528011.html
  [5]: https://www.octf.gov.uk/OCTF/media/OCTF/images/publications/Cross%20Border%20Organised%20Crime%20Assessment/Threat-Assessment-Report_Layout-1.pdf?ext=.pdf
  [6]: https://www.azom.com/article.aspx?ArticleID=13244