There was a lot of racial tension during the war. Black soldiers were not treated like white soldiers were, even though they wanted to fight in the war. They were segregated (they were desegregated in 1948 by President Truman) from the whites and given menial roles even though they put their lives on the line for the United States. They also received less medals than white soldiers did. When these soldiers came home they still faced a lot of racial tension.
Postwar did bring a massive civil rights movement to the forefront, which eventually led to equal rights. Of course many people did not like the fact that blacks were demanding equal rights. There were many people who believed that blacks are not equal to whites and there were people both black and white who fought together for equal rights.
During WWII, racial tensions in the US increased to some extent. This was largely because black people were moving to places they hadn't been before (cities in the North) and because some whites in the military were being exposed to "new" minorities (like in the Zoot Suit Riots).
You can say that the war ended up having a good effect on race relations, though. Because of our anti-racist rhetoric that we used against Nazi Germany, it was somewhat easier for blacks to start to demand rights in the US. This helped lead to the civil rights movement.
There was rampant racial tension during World War II in America, as exemplified by a number of incidents between the whites and people of the minority groups, including African Americans and Mexicans, among others.
The Sleepy Lagoon Trial was one such incident. Seventeen Mexican American youths were accused and shortly convicted of murdering a farm worker. The case was marred by racial prejudice from both the prosecution and the presiding judge. The situation earned sympathy for the defendants by the general public, and when the case came up for appeal, the conviction was overturned and defendants acquitted by the appellate court.
Another racial incident was the Zoot Suit Riots, where white soldiers, marines, and sailors attacked Mexican American youths identified by the “Zoot suit” style of dressing (high-waisted trousers worn with long coats). The white servicemen accused the Mexican American teenagers of being unpatriotic and displaying a high level of extravagance during wartime.