Is there a halachic reason not to mention fallen IDF soldiers names on shabbat or yom tov?

Shalom U'vracha, You shouldn't read or speak about the holy soldiers that had fallen if that makes you sad. This is because on Shabbos we aren't allowed to talk about sad issues (Mishna Brura 307:3, Shmirat Shabbat Kihilchata 29:50). However, if you want to pray for the ilui nismat (elevation of the soul) of the soldiers that fell, that is a mitzvah! (see Rema 284;7) All the best!