I am currently creating a country located in a place which experiences only subarctic/arctic climates in the real world. I would like to know how the region's climate could be artificially changed without causing change to the rest of the world. For example, I am thinking about a giant geyser powered by the Earth's core which generates massive heat. If that is not physically possible, what could be the other solutions?
All sorts of things cause local climates to vary (sometimes quite sharply) from their immediate surroundings. If you're wanting to make subarctic/arctic into tropical? That's going to be extremely difficult without some way to isolate airflow.
If you're not warming it beyond cold-temperate, then a body of water located centrally and deep enough to be warmed by near-surface volcanic activity is a good start.
Physically speaking, any energy (heat) present in the area will eventually move into the surroundings, causing changes in the global environment. If the area in question is kept smaller, that effect will be lost in the background noise of the rest of the climate. But if the region gets quite large, then you're going to have noticeable impacts on regional and possibly global scales.
Artificial methods to accomplish this would involve mountain sculpting to impact rainfall/prevailing winds so that airflow exchange is reduced, the ability to change color scheme on a day/night cycle so that you get good absorption during the day but reduce radiative losses at night. Maintaining a high moisture content in the area will help retain heat once it's been warmed up in the first place.
And, of course, any form of heating system can keep an area warm. There will be losses, but it's doable with enough energy - air-source heat pumps are a good mechanism because the energy you're loosing into the air surrounding the region is heat you took away from it in the first place.
All of this will require levels of understanding of climate dynamics beyond what I'm aware of us having, but at a theory level it's possible.
A MUCH simpler solution, however, would be an enclosure. A good enclosure isolates the airflow and powerfully reduces energy losses and energy demands.
Bring the ocean
You will make your land coastal, by excavating and moving away lands that lie between your country and the ocean coast. The presence of large bodies of water in proximity to the land moderates the climate - this can be seen with Britain which is warmer than inland places at comparable latitide as well as San Diego which is cooler than sites only a little ways inland.