What tensions emerged between 1945 and 1975 to challenge the three-world order? What challenges did they present for various states?

The "three-world order" was a post-World War II model to organize the world. The First World was the United States and its allies, the Second World consisted of the Soviet Union as well as various smaller nations, and the Third World was essentially the rest of the globe. The First World was characterized by its practice of democracy and capitalism. The Second World was, alternatively, anti-capitalist and communist and constructed new systems of governance based on the failures of capitalism. Both the First and Second Worlds had distinct, internal tensions as well as external tensions with each other.
To begin, let us discuss the internal tensions in the First World. While capitalism brought wealth and the growth of the American middle class in the 1950s, nothing was done to calm the social, racial, and cultural issues still ingrained the country. In the 1960s, women and racial minorities were denied what are now considered basic human rights. This issue contradicted the image that America was projecting to the rest of the world: that American capitalism afforded everyone the highest standard of life possible. Anti-capitalists and communist supporters used this incongruity to argue that it was, in fact, the Second World (i.e. the communist world) that provided a better life (the height of these tensions reached their peak in the United States during the Vietnam War, a conflict that involved all three Worlds).
Therefore, a challenge to the First World order was in preserving the idea of the "American Dream" when racial injustice and the poor treatment of minority groups made the ideals of the American Dream (e.g., democracy, freedom, and capitalism) seem hypocritical.
Concomitantly, the Second World experienced tensions of their own. Abhorrent human rights violations committed by Second World powers were consistently suppressed by state-run media, and consequently the citizens continued to suffer. The promises of communism proved empty when basic living conditions were denied to citizens. Additionally, major tensions arose when China and the Soviet Union (both significant powers of the Second World) broke off negotiations in 1960. Starvation and extreme poverty had sparked peasant revolts that threatened the Second World as a whole, forcing China to back out of negotiations.
Much like the First World, the hypocrisy that was reveled between governing ideals and material reality presented a challenge to the Second World order. Despite the fact that communism promised the equal treatment, socially and economically, of all citizens, the reality under brutal dictatorships resulted in extremely poor living conditions, widespread poverty, and general discontent with these oppressive systems of governance.
One of the most notable challenges in the Third World (which consisted of all other entities not included in the First- or Second World) was unique to countries that experienced the withdrawal of European powers following the process of decolonization. Despite the First World and Second World powers experiencing major problems, at the very least they had established systems of government and leadership. After the withdrawal of European governments from colonized countries, the resulting power struggle made the Third World vulnerable to dictatorship, a struggle that continues to this day. This process was not made easer by the interference on the part of First- and Second World powers that sought to exploit the natural resources in countries that were attempting to establish independent governments.