The negatives of interracial relationships are many. (I am interracial myself and married to somone who is not the same race as me, so this should not be construed as a racist sentiment from someone who is against such things.)
For the couple, an interracial relationship can be more stressful. It is not the norm and so people are more likely to be critical of such relationships. It puts little, seemingly minor stresses on people. For example, having people assume that your kids are not yours because they don't look like you on a racial basis is annoying. There is also the possibility of one or both families being less than accepting of the relationship. Such things can cause difficulties that would not be present in a single-race relationship.
If the couple has children, they can face problems as well. They may not be accepted by other children of either race. This may cause them to be unhappy because they cannot fit in with either race. This is bad for both the children and the parents.
These sorts of challenges are the major con with respect to interracial relationships.