Question : As an expat work permit holder living in your City, do I need an exit permit to leave the country for a holiday?

When you first enter Ecuador, you will receive a T-3 tourist stamp in your passport, it allows you to stay 90 days in Ecuador.
If you wish to stay more than 90 days within a year, It is necessary to apply for a visa called 12-ix, it can get once you are in the country at Ecuador consulate and it allows to stay for up 180 days in a year, often referred to as the Tourist, Commercial or Sports Visa.
If you are planning to apply for permanent residency, you are required to have the 12-IX visa when you file your application.