Is the increased/decreased enzyme activity (tyrosinase) caused by an environmental factor (UV radiation) considered to be gene expression?
Unless there's a feedback mechanism shutting off gene expression (which controls the proteins made by a cell), activation or deactivation of an enzyme is not in itself a matter of gene expression, it's a matter of external factors controlling the enzyme's activity. You need to decide on what exactly causes skin pigmentation: the enzyme or cell proliferation, and what is under the control of gene expression and what isn't.

Most of what you find on ultraviolet light and melanin will be somehow related to melanomas, the formation of which does involve gene expression, so it's important to keep the two separated.

Ultraviolet light does cause proliferation of melanocytes, which cause the skin to darken. Melanocytes impart melanin to keratinocytes; these cells then are what give you a tan. The keratinocytes undergo constant turnover, so are eventually shed, and in the absence of continued exposure to UV radiation, there is no need to keep producing melanin-heavy keratinocytes. 



  The keratins and proteins within the stratum corneum act mainly by scattering and absorbing the UV. UV-R sets in action an integrated mechanism for the formation and delivery of melanin within melanosomes from melanocytes to keratinocytes. This mechanism is probably triggered by keratinocytes, which respond to UV-R with bursts of mitoses and with increased production of ET-1 and POMC, thus creating a new demand for melanosomes. The mitotic rate of basal keratinocytes increases a day after UV exposure, reaches a maximum 2 days later, and maintains this level for ∼1 wk. It then declines, and the skin regains its original thickness after 1–2 months if there has been no subsequent exposure.


Continued exposure to UV light stimulates the melanocytes to grow. This increases the chance of the development of melanomas. Since skin cells containing melanin are shed (there's no getting around that), in light-skinned individuals, keeping a tan is to keep injuring the skin and risking carcinogenesis.

Melanin Pigmentation in Mammalian Skin and Its Hormonal Regulation
Human skin pigmentation: melanocytes modulate skin color in response to stress
Mechanism of UV-related carcinogenesis and its contribution to nevi/melanoma