Will there be customs and immigration controls between Northern Ireland and the Republic of Ireland after Brexit?
It's possible but unlikely. Both the UK and European Council have in their guidelines for Brexit negotiations recognised the special position between the UK and Republic of Ireland. The European Council called for "flexible and imaginative solutions", including "the aim of avoiding a hard border, while respecting the integrity of the Union legal order".
Theresa May, in her letter triggering Brexit, wrote:
[W]e must pay attention to the UK’s unique relationship with the Republic of Ireland and the importance of the peace process in Northern Ireland. The Republic of Ireland is the only EU member state with a land border with the United Kingdom. We want to avoid a return to a hard border between our two countries, to be able to maintain the Common Travel Area between us, and to make sure that the UK’s withdrawal from the EU does not harm the Republic of Ireland. We also have an important responsibility to make sure that nothing is done to jeopardise the peace process in Northern Ireland, and to continue to uphold the Belfast Agreement.