Is global warming a threat to your mineral collection? Maybe.
First, let's talk mineral stability. Many mineral species are described that are not stable at room temperature, pressure, and humidity. However, most are stable in the environments in which they form (some are formed metastably, which won't be discussed more in this blog entry).
To form, minerals need a supply of nutrients (elements), and the appropriate pressure and temperature ranges. The conditions needed for mineral stability also include the relative amount of water available, relative acidity, and oxidation state. Minerals that contain other elements that may be in the gas or fluid phase, such as sulfur, selenium, and fluorine, may have additional requirements for stability.
Now look at minerals that are outside their stability range. Will these immediately decompose? Not necessarily. If the amount of energy needed to start the change or decomposition is high, the mineral may stay in the same state despite adverse conditions. This is called a metastable condition, and the best example is diamond.
However, many minerals do break down, and the conditions on earth that form minerals may not be represented in your collection. The most obvious low-temperature example is ice; another is antarcticite. Neither of these are stable at 20-25 C.
The opposite case is for high-temperature minerals whose crystal structures spontaneously distort at lower temperatures. Probably all so-called specimens of "beta quartz" are now alpha quartz, unless there is a mineral collection that keeps specimens above 550 C.