Brexit - some tax implications for Irish workers and businesses trading in the UK and UK workers living in Ireland
For Irish workers living in the UK, there is likely to be some sort of free movement of people between the 2 islands just as there has been from the foundation of Irish state. There may be more formalities in terms of passport control, because someone leaving Ireland and entering the UK will be also leaving the EU and going to a non-EU country.
In terms of someone who is an Irish citizen working in the UK, the tax situation will not change. If you are employed in the UK, then you pay payroll taxes there and that income, as it is a salary-type income, is not taxed in Ireland in any way shape or form.
Similarly UK workers in Ireland will be taxed on their salaries in Ireland and there will be no further tax implications.
For Irish companies selling in to the UK, there is no VAT on sales of goods as long as the goods are exported. That is the same for any goods exported outside of the EU.
For self-employed business people who have a limited company, residence of that company is easier to establish in a given country, than it is for an individual to establish residence. So for example if you are in Ireland and have a UK-registered company, that UK registered company will remain resident in the UK as long as there is a registered office there and all important decisions about the direction of the company are taken in the UK. The same applies to a company which is registered in Ireland where the directors work in the UK, that company can be resident in Ireland even though the director may spend most of their time in the UK.