With some exceptions (one being red blood cells), all cells in the body contain the complete genome of the individual. The expression of the genes however is, as mentioned in the comments, heavily regulated and different in various tissues. Early efforts at gene therapy have had mixed results, including some fatalities, and the field is still highly experimental.
Depending on the trait you want to change, you might have to manipulate the genes and/or gene expression in just a particular tissue, or in several. The rates of cell death and replenismhent varies between tissues, and will affect the stability of any introduced mutation. Some traits may depend on too many genes to effectively manipulate with current technology, and eye color does depend on several genes. From the Wikipedia page on Eye color:
Eye color is an inherited trait influenced by more than one gene.
These genes are sought using associations to small changes in the
genes themselves and in neighboring genes. These changes are known as
single-nucleotide polymorphisms or SNPs. (...) There is evidence that
as many as 16 different genes could be responsible for eye color in
humans; however, the main two genes associated with eye color
variation are OCA2 and HERC2, and both are localized in Chromosome 15.
So depending on the original and desired color, you may or may not get away with changing a single gene. However, the pigments that determine eye color are produced in two separate tissues in the eye, making matters a bit more complicated:
In humans, the pigmentation of the iris varies from light brown to
black, depending on the concentration of melanin in the iris pigment
epithelium (located on the back of the iris), the melanin content
within the iris stroma (located at the front of the iris), and the
cellular density of the stroma
Note that cellular density also plays a role, and this presents a much harder property to change with gene therapy, if it is at all possible.
Eye diseases have been a popular field for gene therapy research and trials lately, and there seems too have been quite some progress. However, the possiblity of using gene therapy for changing eye color has understandably not received much attention, and so there are few if any papers to consult. What can be said though, is that eye color is a complex trait, with a partially known genetic basis. It is plausible that gene therapy can be used to change pigment production and thus eye color, but even so there would likely to be many limitations. on the changes that can be made.