Yes, I think that other EU countries will follow in Britain's footsteps by holding votes to leave the EU. The Netherlands and France are two countries that may do so as well. The people of both countries are unhappy about the bureaucracy of the EU and the influx of immigration, and Britain's actions may encourage others to follow suit.
Some EU countries are likely to hold their own referendums in light of the UK's Brexit vote to leave. Many countries in the EU are dissatisfied with the organization's economic and political structure. Many members want to have more control over their banking systems, borders, immigration, as well as other important issues. Therefore, it is likely that there will be other countries that vote to leave the EU as well.
The necessity of having a vote to see where your citizens lie is more than just a given. It will happen in many Democratic-ish nations, and it is only a matter of time before they are held. The true question, however, is how the nations will vote. And based on the BREXIT/REGREXIT issues to date, the nations will almost certainly vote to stay within the EU.
The other countries don't have the money/wealth or resources and time required to leave the European Union. The EU provides too much to those countries who may not be able to suport themselves without havin a VERY strong connection with another group of nations (such as the EU). It is an entity that could be considered as a double edged sword.