It is forbidden to remove tzitzis strings from a garment unless one intends to affix them to another garment. If, however, the garment is no longer able to be used for the mitzvah (e.g. if a corner was cut off, if it is being sold to a non-Jew or if it is falling apart), removing the strings is permitted. If the owner wishes to replace the strings with nicer tzitzis, they can also be removed. The poskim disagree about whether old strings may be torn and rendered non-kosher during their removal or whether they must be removed in their entirety.
[שו"ע טו, א, ומשנ"ב ג]