Global citizenship is in itself a controversial issue, not because of inherent negative aspects, but because most people are nationalistic in viewing their own nations versus the rest of the world. Global citizenship implies that one is not a citizen of any one country, but of all countries together, and that can be problematic for people who view their country or culture as superior.
One problem that many have with the concept of global citizenship is the increase in scope and power that a world government would necessarily need. In order to protect its citizens and enforce laws, a global government would be enormous both in its size -- needed to create, interpret, and implement laws for all countries -- and its power over the citizen. This government would also need to impose enormous taxes to fund itself and its endeavors. For those who believe that government should be restrained in its power and scope, this solution would be all-but impossible to sell.
Another problem is that not every country will always go along with a world-citizen view; many countries have extremely strict immigration and naturalization laws, and they might ignore rights granted under global citizenship laws enacted outside their country. This speaks to the previous point, requiring an enormous government to govern both the people and the individual countries.
Finally, the concept of national pride could become obsolete, and that is a problem for countries with a rich national heritage. Italy, for example, has a history that stretches back thousands of years, and the Italian people might not be grateful to a system that essentially places their people on a level scope with all other nations. The loss of national identity is something that most countries fight against; even "global citizens" with multiple citizenships often view one specific country as "theirs," unconsciously practicing nationalism even if claiming otherwise.