All citizens of the world who wish to have a temporary permit with labor ties in the Dominican Republic can apply for a temporary worker visa at the consular missions (embassies or consulates).
This is the model of work permit granted by the country.

Temporary Workers: 

It is the Non-Resident permit for a foreigner who enters the Dominican Republic with a work contract, endorsed by a company legally constituted in the country. The validity of this permit is for one (1) year, and may be renewed for one (1) more year, if proof of the extension or renewal of the work contract is provided.

Requirements:

1. Completed temporary worker visa application form. Download here.
2. Passport valid for at least six (6) months. According to the application regulation 631-11 and a complete copy of the passport.
3. Temporary Worker Visa (VTT).
4. Inextensive Birth Certificate, apostilled or legalized, as appropriate.
5. Two (2) recent photographs, front view, size 2×2, white background.
6. Certificate of No Criminal Record from your country of origin or the country where you have resided in the last five (5) years.
7. Labor Contract and Resolution of the Ministry of Labor that orders the registration of such contract.
8. Constitutive documents of the company, copy of the RNC, Certification of Internal Taxes and any other document that the DGM considers necessary.
9. Receipt of the medical examination, approved by the General Directorate of Migration.
10. Contact Information:

• Address: Autopista 30 de Mayo, Esquina Héroes De Luperon, Santo Domingo Dominican Republic.
• Phone: +1 809 508 2555 Ext. 2104 / 2018.
• Email: noresidentes@migracion.gob.do

Procedure:

1- Enter the DGM service portal, here.

2- Register your username and password in the system (If you have already registered previously, enter your previous username and password).

3- Select the service:

• Choose the option "APPLICATION LIST".
• In the "AVAILABLE SERVICES" option, select the LOCATION-OFFICE where you will receive the service.
• From the "LIST OF SERVICES", choose the service to be requested.
• Complete all the information required in the "INFORMATION QUESTIONNAIRE FOR APPLICANT".

4- The scanned documents must be in JPG format for the web application.

5- The medical examination must be performed within three (3) days after payment (including the day of payment), otherwise you must pay again. You must wait (10) ten (10) days to know the results of these.

6- All documents deposited at the DGM must be originals and must be notarized, legalized and apostilled, as applicable.

7- Hours of service: M-F | 8:00 a.m. - 4:00 p.m.

8- Fees: File deposit - RD$5,000.00; Medical exams - RD$4,500.00 ; Loss of receipt - RD$500.00.

9- Completion time: 90 calendar days.

10- To complete the whole process register in the portal of the government of the Dominican Republic, here https://personal.migracion.gob.do/ and follow the steps.

More details here:

Other related permits:

Short-term permit (pcp): 

Special short-term stay permit for a member of a group, due to its activity, that will stay in the Dominican Republic for a period of more than two (2) months and up to eleven (11) months, guaranteed by a Dominican institution or company. This permit is non-renewable.

Foreigner member of a group, due to his declared activity, such as: business, volunteer community service, short term student, artist, athlete, religious.

Requirements:

1. Passport valid for at least six (6) months. According to the application regulation 631-11.
2. Valid Multiple Business Visa (NM) or Student Visa (E).
3. Copy of the air ticket to return to your country.
4. Four (4) recent photographs, of the same set: two (2) front and two (2) right profile, size 2×2, white background.
5. Updated certification of the institution or company that guarantees the stay of the foreigner.
6. Medical certificate.
7. Medical insurance.
8. Documents that endorse the economic solvency.
9. Register in the portal and do the same registration steps as mentioned above, here.
10. Costs: File deposit - RD$5,000.00; Payment of stay - according to table of payment of stay; Payment of expired visa penalty - according to table of payment of expired visa penalty.
11. Time to receive a response: seven working days. 
12. The process is done in person.

More information here.