Question: If you omit the italicized words in Psalm 3:8 in the KJV, the meaning seems somewhat different. Is there something in the Hebrew to support those words being italicized?
Answer: The funny thing about those "italicized" words in the KJV, which are identified by being italicized, is that in my experience, they are not very consistent. Yes, sometimes those italicized words were added, but at other times they are not. In the case of Psalm 3:8 two words are italicized, belonging and is.
The first part of that verse is ליהוה הישועה (l'YHWH ha'yeshu'ah). The word ליהוה (l'YHWH) is the name YHWH with the lamed (ל) as a prefix, which means "to" or "for," but can also mean "belonging to." The word הישועה (ha'yeshu'ah) means "the salvation." So this phrase can be translated as "to YHWH is the salvation" or "The salvation belongs to YHWH." As you can see, the KJV is correct. In this case the translators have italicized the word "belonging," because it the Hebrew only says "to YHWH." But look at Psalm 94:1, which reads, "O LORD God, to whom vengeance belongeth; O God, to whom vengeance belongeth, shew thyself." Notice that the word "belongeth" appears twice and both times it is NOT italicized. The Hebrew literally translates as, "God vengeance YHWH, God vengeance shine." But we can translate this as "God of vengeance-YHWH, God of vengeance-shine." Notice that there is no such idea of "belonging" in this verse and the KJV "added" this word, but it is not in italicized.
Now let's take a look at the second italicized word in Psalm 3:8, which is the word "is." Hebrew has no word for the verb "to be" or its conjugations; is, was, were, are, etc. So we have to add these in in the English translations. As an example, in Hebrew you would say הוא פה (hu po), which literally means "he here," but in English we would say "he is here." There is however a Hebrew verb that can be translated as "is" and that is the verb היה (H.Y.H) which literally means "to exist," but we would often translate this verb as "is," "was," "are," etc. But again, the KJV is not always consistent. Notice that Genesis 2:9 reads, in part, "grow every tree that is pleasant to the sight." The word "is" is not italicized, but there is no Hebrew word in that sentence that means "is," so it should be italicized, but it is not.