They stop being used, and the person who ascends the throne uses instead the arms of the country over which he is ruling, both as his own arms and those of the state. For example, the present Elizabeth II had arms that she used before she ascended the throne that she does not use now. Likewise with King Albert II of Belgium and Queen Beatrix of the Netherlands.
As regards William the Lion, it's not known. The arms of Scotland in their present form did not appear until several generations later with King Alexander III. It's likely he used a Lion in some form, but what is doubly interesting is that the male-line descendants of one of King William's illegitimate sons did not use the royal arms of Scotland in any differenced form but instead used the arms of the family of the wife of this illegitimate son, much later they adopted a version of the Royal Arms of Scotland, differenced for illegitimacy. Which suggests that William probably didn't bear arms.