Could I lose my current work permit if I apply for a Greek Golden Visa?

Yes, you can have only one resident permit in Greece.

If your work permit is valid when you apply and get your Golden Visa, then nothing will happen to your work permit. Getting a Golden Visa will not jeopardize your existing work permit.

No, a Golden Visa holder can't work in Greece but he can set up a company as a shareholder or be a CEO. According to the the type of your resident permit, you can apply for a family reunification.

According to the current immigration law, every citizen of a third country can reside legally in the country upon holding one legitimate title of residency. From the other hand, in no case does the resident permit for property owners provide access to any type of employment. Employment, according to paragraph 6, article 20, of Law 4251/2014, does not include the exercise of economic activity in the capacity of a shareholder or a chief executive officer for an already existing company. So, according to the above data ,the alternative options for you are the following: to proceed to Golden visa issuance, by property acquisition and at the same time your family members are granted for the Golden Visa privileges too. In this case you lose your working permit but have the chance to exercise economic activity as above. Second, your wife could proceed with the Golden Visa program as a main investor, and you won’t apply for the same program as family member by keeping your existing work permit.

Here is some additional information to my previous answer. There is a third option to your question, the case of family reunification under the terms and conditions and besides the Golden Visa program. Refer to Article 69, the requirements for family reunification.