Because it's not entirely a genealogy
Korah also appears in 1 Chron 6:38, so it is unlikely this his rebellion in Numbers 16 was the cause of him being omitted from a later chapter in 1 Chron.
The one "son of Izhar" mentioned in your verse, 23:18, is Shelomith, who is not among Izhar's sons listed in Exodus 6:21, Korah, Nepheg, and Zichri. It's not only Korah missing, but all three. The same happens with the sons of Hebron (1 Chron 2:43 cf 23:19). So, something more must be going on.
Using Bible to interpret Bible, consider the same chapter...
23:6 the opening states that David organized them by sons of Levi, not that this is explicitly a genealogy.
23:24 the closing states that the previous lists were heads of houses and twenty years older who did the "work service of the house of the Lord". Again, not that it was a genealogy. More importantly, they were alive at the time; Korah was long dead because he lived in the time of Moses, not David. (It is likely that Shelomith was David's contemporary, based on the opening and closing verses listing adult contemporary workers organized under David.)
Also, the wording of 23:18 uses the plural "sons", but only one son who is "chief", again bolstering the vv6, 24 interpretation (above).
So, his name was probably not omitted because of his rebellion because it wasn't omitted at all. The text is simply listing who was in charge of that part of Levi's family.
Note: Genealogies were "allowed" to omit generations because they were often given to make a point. They also had understandable discrepancies. A classic example is the Gospels. Luke 3:33 records two generations between Amminadab and Hezron (Admin & Arni), but Matthew 1:3-4 only records one between them (Ram AKA Aram). Two things: 1. Arni or Admin could have been Aram with all the language confusion of names, just as Paul went by different names—Hebrew and Greek—, and these are Hebrew namess recorded by Luke—a Gentile writing in Greek. 2. Matthew was a Jew had a Jewish focus, making it all the more likely he would omit a name in a genealogy. Again, in those ancient times, "ancient" was "contemporary" to them, and they didn't find the same value we do in being exhaustive.