No, probably not.
To enter the Dominican Republic you are required to get a visa only if you are from a country listed as requiring as such. See their list here (in Spanish only). It also includes a list of countries that do not require neither tourist card or visa and can enter with only a passport. In case you are on the visa-requirement list you will need to fill out a visa application here (pdf file).
However, in general all countries that are part of the European Union as well as the UK and North American countries (US, Canada) do not require a visa. They only require you to bring your passport. Upon arrival to the country (either on the plane or the airport) you will be asked to fill out a small immigration form.
Upon passport control you will have to pay a tourist tax of $10US. You will then receive a tourist card and will be able to stay in the country for 30 days, but no longer than 60. If you overstay these 60 days you will be required to pay a hefty fee upon departure.
Tourist Card Sample
To reside in the country the requirement are somewhat different. See here for the steps on that.