Yes, the relationship between Britain and its neighboring countries will suffer as a result of Brexit. As the fifth biggest economy in the world, Britain is a very important part of the European Union. Its exit is likely to cause turmoil across the EU, leaving the other countries of Western Europe scrambling. Countries such as France and Germany will likely impose tariffs on Britain, leading to souring relationships between the countries.
As for how a Brexit would affect the EU, the impact would be widespread and drawn-out. The actual process of the U.K. leaving the partnership and establishing new agreements with remaining EU countries would take roughly 10 years. As Great Britain forged new contracts with the nations left in the consortium, many businesses would face substantial uncertainty.
Great Britain's relations with Europe will suffer because of the "Brexit" vote. Many European Union members now feel frustrated with the U.K. for voting to leave. The remaining countries in the EU want to make Britain's exit as painful as possible to serve as a warning to other members that might be considering leaving as well. This will likely strain relations between the U.K. and its neighbors in Europe.
Even with Britain deciding to leave the EU, positive political relations with neighboring countries are to everyone's advantage. Britain is a large, wealthy economy and a good trade partner. EU member countries will simply have to renegotiate trade terms, but once the period of change happens, I believe relations will continue to function normally for all neighboring countries.