Yes, your EU driver's license is generally valid in other EU countries, even if you become a resident. There a few exceptions/caveats:
- It's only true if you passed the exam in an EU country to begin with. If you obtained your EU license in exchange for a non-EU license, then it might not be recognized by other EU countries. They will know about these types of exchanges based on the information codes (section 12), which are visible on the license. For example
70.0123456789.NL means that your license was originally from the Netherlands.
- If the local license has a limited duration (“administrative validity”), you might need to exchange your foreign license after some time (I already discussed this in If I obtain a driver's licence in Germany as a non-German EU-citizen, would I have to revalidate it after leaving Germany?). Even in this case, you can always use it for a few months (not sure what the exact duration currently is, it used to be one year I think) after moving.
- All this is valid for a category B license for regular private cars. There are still some small differences with the categories (tractors, motorcycles, etc.) and I don't know all the details. I know people who have lost or gained a category by exchanging their license.
- You can drive with your license but not necessarily use it for other purposes. For example, in the Netherlands, you can use a local driver's license as ID at your bank but a bank employee once told me they wouldn't accept a foreign license for that. Since they are not a public administration or anything, I assume they are within their rights to do this.
In any case, if you need to get a local license (because your original license is expiring, it was lost or stolen, etc.), it's not that expensive and doesn't need to be done each year. There might be some paperwork and a few annoying formalities but in my experience for a regular B license, it will cost you around 50-100 € for a ten-year or fifteen-year license.
The EU provides some information about that on europa.eu