You are allowed to remove tsitsis from a begged in the following cases:
The owner died
Tallis wore out
To replace them with nicer tsitsis because of hiddur mitzva
To replace the old ones with new ones
If one of the eight strings snapped and is now only keday aniva (4cm) and you wish to replace them with fully intact tsitsis (MB3)
If it is a bother to untie and unwind the tsitsios, you are allowed to just cut them off – it is not considered bal tashchis. (MB3)
There is a custom to stitch on to each corner a whole cloth measuring three by three finger breadths. (7.2cm x 7.2cm) (MB16)
If the hole ripped but not all the way to the edge, stitch it with a thread that is not the same material as the begged. Better yet, use a thread of a color other than white. (ShA5 – MB18)
If it tore all the way, attach another piece of cloth measuring 3×3 fingerbreadths onto the corner and hang the tsitsis onto that piece. (ShA5 – MB21)