Is is possible? Yes.
Technology-wise, you would do the same thing you'd do for language localisation (see this related SE question for the specificities).
Localization refers to the adaptation of a product, application or
document content to meet the language, cultural and other requirements
of a specific target market (a locale). Source
Seems like the definition fits the icon situation perfectly. W3C actually mentions "Symbols, icons and colors" in their list of customisable elements. Just as it happens with content, images are loaded with cultural baggage.
The apple developers site has some guidelines in respect to internationalisation too:
Create or modify language-specific versions of nib files, text, icons
and graphics (especially those containing culture-specific images),
audio, and video for each locale. Source
Now, in regards to the cases, that's a tricky one because it's not easy to hunt down some good examples, but take a look at this:
In Bangladesh, Iran, and Thailand the "thumbs up" gesture (facebook's "Like") is traditionally an obscene gesture, equivalent to the use of the middle finger in the Western world. Apparently nothing happened, but I bet it was something FB's developers and designers had to consider.
Also, the example from UX.SE cited in the comments has a nice answer (too bad the Q was closed).
Animal symbols can also be dangerous. For example, owls symbolize
wisdom in the United States, and an e-learning website may use an icon
of an owl to symbolize that a user or student is performing well in an
online course. However, owls symbolize stupidity in some parts of
Asia, and Asian students may be insulted, not encouraged, by such an
Religious symbols can, of course, be particularly sensitive.
Microsoft’s geopolitical product strategy team once avoided
embarrassment by preventing the release of the company’s Office XP
software containing a moon and stars astrology icon that resembled the
Islamic Hila symbol. When religious symbols cannot be avoided, they
must be localized, such as when the Red Cross has been adapted as the
Red Crescent in the Middle East. Source
One thing is sure: This is a job you can't do based on assumptions. Feedback or research is crucial.
I found this interesting discussion in meta:
And some nice related articles: