If the Underground Railroad really was a threat to sectional harmony, it was a threat because it helped slaves escape from the South. This angered Southerners and made Northerners feel more negatively towards the South.
Southerners would have felt that the Underground Railroad was an illegal organization meant to deprive them of their rightful property. After all, slavery was legal in the United States and their slaves were legally their property. The Underground Railroad, then, was an organization (of sorts) whose only purpose was to make slaveowners lose property that was legally theirs. In a sense, this was like an organized crime ring that was stealing from the people who owned slaves. The Southerners would have been very angry that Northerners were willing to help steal their property and they would have been angry that Northerners approved of the Underground Railroad.
From the Northern point of view, the Southern reaction to the Underground Railroad would have shown the evils of slavery. Many Northerners were not really against slavery, but they did seem to believe that it was understandable that slaves would want to escape. As we see in the reaction to the Fugitive Slave Act of 1850, Northerners were angered at the idea that slaves who had escaped to the North could be taken back into slavery. It seemed heartless and evil to do this. Therefore, when the Southerners got mad about the Underground Railroad, it made them look worse in the eyes of the North.
In this way, the presence of the Underground Railroad caused (at least arguably) greater tensions between the North and the South.