The main positive aspect of global citizenship is that those people who see themselves as global citizens will have a more holistic view of the world. They will not think first of their own country but will think of what is good for the world as a whole. This might, for example, lead them to support efforts to fight climate change even if their own country is harmed economically in the short run. Global citizenship can, then, promote policies that are better for the world.
The major negative aspect of global citizenship is that it can lead to the "good guys" being harmed and the "bad guys" being rewarded. If the good guys act as stated above, those who refuse to act as global citizens will be rewarded. They will benefit by gaining economically and they will benefit from the reduction in climate change brought about by others' sacrifices.
Thus, global citizenship can help bring about positive policies, but it can also hurt those who act in the right ways and help those who do not.
The concept of global citizenship encourages people to see themselves as being citizens of the entire world and understanding the interdependence of all living creature in our world rather than identifying simply as a member of a tribe or nation. In many ways, this is extremely positive. It leads us to understand that things like environmental degradation and climate change affect all of us and that, for example, we cannot ignore human rights abuses in the sweat shops in Asia that make clothing or deforestation in Brazil.
On the negative side, it can induce a sort of apathy or inaction. Most of us cannot have a major impact on the entire world, but we can affect what goes on in our own communities. Thus it is important to "Think globally, but act locally." In other words, if you are concerned about climate change, simply expressing outrage on social media about the global problem doesn't do much to solve it, but fighting to have solar panels installed on the roofs of all school buildings in your town is a small step that can make a big difference. You cannot end global poverty, but you can volunteer at a local food bank. You may not be able to end the oppression of women in the Middle East, but you can donate gently used clothing or free samples of personal care products to a local women's shelter.
In other words, although it is good to think in general terms as a global citizens, when it comes to actually solving problems, it's important to think and work locally.