Imagine that beak color in a finch species is controlled by a single gene. You mate a finch homozygous for?

A incomplete dominance ( heterozygous individuals have an intermediate, or in-between, phenotype).

incomplete dominance results in a new phenotypic expression not seen before in the offspring  think red plus white give you pink flowers.    in co-dominance both alleles are expressed simultaneously...think red and white flowers give you red and white striped flowers.  with your discription of ivory crossed wtih orange giving you a pale ivory -orange bill it is hard to assertain from the description of the bird's beak whether this is a case of co-domiance or incomplete domiance.  it is not crossing over, pleitropy(where one gene affects many physical traits) nor polygenic inheritance (where many genes affect one physical trait)