The Story
Dating one's first-cousin is a bad idea. First-cousins that marry and have children are committing incest; those children will be born severely deformed.
It is so bad that in many states, first-cousin marriages are not only illegal, they are considered criminal offences.
Facts
Counter-example
Queen Cleopatra of Egypt is an extreme counter-example to this idea. All of her ancestors, for five generations back were descendants of the same couple. Most people have 64 great-great-great-great-grandparents; she had two.
Laws
It is true that some U.S. states consider this a criminal offence. Cousin marriages are also forbidden in China and a few other countries. But in most of the world, it is legal and often quite common.
Inheritance
People's DNA has two copies of each gene. If one copy happens to be non-functional, that's usually okay as the other will provide whatever genetic information is necessary.
The problem is that if a common grandparent has a defective gene, two grandchildren could each inherit that gene and both could pass it on to their child. If that child has a gene where both copies are defective, the child will have the disease, disability, deformity, etc. associated with it. If the grandparent does have a defective gene, the chance that a great-grandchild will inherit two copies of it through their first-cousin parents is one in sixteen (6.25%).
Consider a common inherited disease such as Cystic Fibrosis. Among northern Europeans, one person in twenty-five carries the damaged gene. This means that the chance that a child of first-cousin parents inherits both damaged genes from its double great-grandfather is 1/16th of 1/25th, or one in 400 (.25%).
Most inherited diseases are far far rarer than one in twenty-five, so that .25% rate for C.F. represents an extreme case.
Risk
Now suppose we consider a rate of .25% to be too high a risk, and use it to justify declaring such marriages illegal. What are the implications of this?
If we applied the same criterion to Down's Syndrome, we would have to make it illegal for women over 32 years old to conceive children, as the percentage of 16-week fetuses with Down's Syndrome is higher than .25% for 33 year old mothers. And perhaps we would have to make it a criminal offence for any woman over 42 years old to conceive, as the chances then are ten times higher than that.
Legally restricting conception based on the age of the mother is far more justifiable than restricting it based on a first-cousin relationship. But no jurisdiction has ever done that.
Why
In the late 1800s, not understanding how genetic inheritance works, some scientists believed that congenital diseases could be weakened by always marrying very unrelated people. Alexander Graham Bell, for instance, strongly advocated banning marriage between deaf people, so that the disease would be watered down rather than concentrated. He even opposed the use of sign language, as it encouraged relationships among the deaf. Studies at asylums tended to support the view that many physical and mental illnesses were caused by first-cousin relationships.
These ideas were soon disproved by further studies and by new understanding of the laws of genetic inheritance. But by then, public opinion and politicians had already decided that they knew better.