Do all diamond and precious gems require breathing holes and why?

'Breathing holes' isn't a common term used by jewelers who set precious stones in metal to make jewelery.

If, however, you mean openings in the setting that provide the stone access to available light from all sides, then, yes: precious gems are best displayed in settings with these openings.

Finally, imitation gemstones are often set with glue, covering all access to light, except from the top.