Short term, not much. Long term, continued upwards pressure on rent and property price.
What ever way Brexit plays out, the significance of London will be reduced for Financial services etc. The people employed in these are the high paying jobs and these are the only people that have some freedom to move. The only question is how great will the hit on London be.
This will play out by large companies first moving their paper offices to EU cities; this is already in progress. In time people will gradually move. This will happen as new projects get started in these companies. A new project X will be setup in Frankfurt/Paris/Dublin rather than London. Some people will move countries, but mainly EU citizens will be recruited.
I’d expect most of these companies and projects to be moved to Frankfurt and Paris. A small percentage of them will move to Dublin. While in overall terms the numbers of people moving to Dublin will be a small percentage of the total, for Dublin it will be significant.
Anyone that frequents this site will know that there are pitifully few properties available to buy in the greater Dublin area as it is. This has been the primary cause of property and rent inflation in recent years. With an additional source of inward migration of people ready, willing and able to live with Dublin prices will see continued upward pressure in the greater Dublin area. This upward pressure should offset any chaos caused by no deal.