You shall keep my statutes. You shall not let your cattle breed with a different kind. You shall not sow your field with two kinds of seed, nor shall you wear a garment of cloth made of two kinds of material.
Leviticus 19:19

This passage says not to cross-breed different kinds of living things. I believe it means across species (eg. cats vs dogs), but the definition of species in this context is debatable. It doesn't directly address gene splicing, but I suspect God probably had that in mind as well. Certainly he knew in 1500 BC that we'd be mucking around in DNA in the 2nd millennium AD.

More importantly than genetic hybrids and mixed fibers, this verse means we aren't to marry unbelievers nor to mix God's religion with paganism.