The most obvious EU influence on UK tax policy is in the field of indirect taxation, and in particular VAT which is an EU-wide tax with set minimum rates. In the event of the UK leaving the single market it is highly unlikely that the UK will abolish VAT (it currently accounts for nearly 20% of the total UK tax take) but it might well make changes to its VAT law that are not currently permitted, such as to extend some of the lower VAT rates to additional types of goods and services, and to introduce further VAT exemptions. The VAT position for businesses supplying or receiving cross-border EU supplies might become more complicated.
Excise duties are the other main indirect taxes governed by EU rules. Tobacco, alcohol and energy are all subject to excise duties and there are agreed minimum rates for each of these, although states are free to set excise duties above the minimum rates. Again the UK may be able to make changes here.
With regards to direct tax, the EU plays a much lesser role and these taxes are principally a matter for each member state. However, there are a number of relevant EU Directives, which are primarily aimed at removing obstacles for businesses operating within the EU. These include the Merger Directive (which applies reliefs for mergers, divisions, transfers of assets and exchanges of shares which take place between companies in different member states), the Parent-Subsidiary Directive (concerned with profit distributions between associated companies) and the Interest and Royalties Directive (prevents withholding taxes on royalty and interest payments). The benefit of EU withholding tax exemptions may be lost to UK companies, but the UK does have a very comprehensive network of double tax treaties, which should lessen the consequences of this. There may be some impact for UK holding companies from the loss of the EU reliefs for group payments, however (eg, in relation to dividends paid from Germany to a UK parent). Future EU tax initiatives, such as the directive on anti-avoidance and the common consolidated corporate tax base, should not now affect the UK.
Post-Brexit, and were the UK also to leave the EEA, the UK would not be bound by EU law restrictions with respect to State Aid. The government would not then be prohibited from using state resources (such as a more favourable tax regime) to provide an advantage to any organisation or business.
Finally stamp duty reserve tax is imposed at the rate of 1.5% on issues of shares and securities to depositary receipt issuers or clearance service services, in certain circumstances. Following European case law, HMRC no longer seeks to impose this charge and so following Brexit, this charge could arguably be levied.
Overall Brexit will have no immediate effect on any tax legislation which has been incorporated into UK law, although the government might be able to amend the law as it wished. Whether it would want to do so is a different question and the extent to which it could will depend upon the nature of any post-Brexit UK/EU relationship.