Let's start with something totally outside The War of the Worlds. Guess what Wells wanted written on his tombstone? "G-- d--- you all: I told you so." For us, that epitaph captures something of the tone of The War of the Worlds. Like that quote, the tone of The War of the Worlds is a little detached and sometimes, surprisingly heartfelt.
The narrator himself notes that he's occasionally detached:
At times I suffer from the strangest sense of detachment from myself and the world about me; I seem to watch it all from the outside, from somewhere inconceivably remote, out of time, out of space, out of the stress and tragedy of it all. (1.7.7)
A lot of the story is even told as if from some bird's-eye view of the situation. For an instance of real detachment – or a literal bird's-eye view – check out the narrator's description of the mass of people moving out of London:
If one could have hung that June morning in a balloon in the blazing blue above London every northward and eastward road running out of the tangled maze of streets would have seemed stippled black with the streaming fugitives, each dot a human agony of terror and physical distress. (1.17.1)
There's a sense in which the narrator is talking about something super distant from himself, as if what he was saying had nothing to do with him and only had to do with other people (that's the "you all" part of the Wells quote).
The narrator treats everything in a slightly detached manner. After all, that's the best way to judge something. And isn't the tone here a little judgmental? We can't quite prove it, but sometimes the narrator's comments seem to be not just observing things from a detached point of view, but judging things. (Maybe that's the "I told you so" part of what Wells wanted on his tombstone.)
Even though the narrator gives us this bird's-eye view over the crowd, he usually lets us know that each of those dots is a thinking, feeling human being – someone who is probably thinking sad thoughts and feeling terribly upset. Of course, there are times when the narrator is pretty unemotional. For instance, when the Martians first kill off Ogilvy, Henderson, and Stent, the narrator doesn't tell us about their pain and sorrow. He just tells us, matter-of-factly, that "it was as if each man were suddenly and momentarily turned to fire" (1.5.14). Similarly, the narrator can pass charred corpses (which he does a lot) without wondering who those people were or what they were feeling.
But there are also times when the narrator lets us know how deeply the Martian invasion is affecting people, as when "each dot" in the crowd is in "agony of terror and physical distress." The final few chapters, in particular, feature a lot of unexpected emotion. Think about the how touched he is by the Martians' wailing in the dead city (2.8), or his uneasiness at the end of the novel over whether the Martians might attack again (2.10). For all his ironic detachment, the narrator occasionally reminds us that each person he mentions (including himself) has some deep feelings.