Even if readers are radically and nearly exclusively committed to the protagonist, there are several ways for the protagonist to "speak after death".
The protagonist's legacy can speak. (This is covered in the answers to "Ways for main character to influence world following their death", linked in Lauren Ipsum's answer. The legacy of a Cause does not seem to be mentioned in those answers, though "What Would Dead Guy Do?" comes close.)
This may take the form of writings, recollections, and the surviving influence on individuals and the society. One technique for doing this is to introduce shorter stories after the death branching off from an earlier action. For example, if the protagonist as an imperial soldier intervened in an injustice by another soldier, a young observer might be moved to recognize that the imperial soldiers are not just bullies and later prevent an uprising (which might incidentally have diverted troops from guarding a border at a very critical time as covered in the main story). Such can act as a kind of extended eulogy or memorial offering, so even a reader focused on the protagonist may appreciate such side stories.
Alternatively, the narrative order may deviate from the temporal order. This includes flashbacks as mentioned in clockwork's answer but even an inverted order could be used where successive narrative sections are set earlier in the protagonist's life. For example, a novel could begin with the funeral and immediate impact of the protagonist's death, proceed to shortly before his death (showing a part of why he was mourned as he was) and show his last words to his wife, then show how the couple came to love each other so, then show how the conflict which proved their love developed, etc., each section ending with a link to a previous time. A fully reversed order would be challenging to write well, but such is an extreme of the temporal reorderings possible.
Another possibility, mentioned in Lauren Ipsen's answer, is that the death is not complete or permanent. This can be a very dangerous method since the reader can easily feel cheated (tricked with inadequate benefit) and lose some degree of suspension of disbelief. This can also make death seem less final and the risk of death less serious. If risk of death is not a common component of the story, it becoming less serious will have limited impact. If the survival reinforces the nature of the protagonist (e.g., tough to kill or crazy prepared), the protagonist's relationship with others (e.g., rescued by insignificant (and so unnoticed) beings, spared by a villain for better sport, rescued by a villain's henchman), or the protagonist's destiny (protected by fate), then this can strengthen the story.
I would also qualify the statement in clockwork's answer that "it has to mean something and it has to be their purpose". There are several ways that a relatively meaningless death can have meaning in the context of the story. Obviously, for a dark tale, pointlessness may be the point. While such may not be popular, it can provide a meaningful story and press the reader to consider the philosophical question of what is the ultimate purpose of life.
Alternatively, the author can hint, possibly even heavily, at a less obvious meaning to the protagonist's death. Subtle hints may even allow various readers to insert their own meanings. One way to hint at such meaning is for the protagonist to have been significantly influenced by a similarly meaningless death; the meaninglessness of the death then becomes part of the protagonist's legacy.
A meaningless death can also be used to display the character of the protagonist (e.g., displaying a sense of humor even in tragedy or raging against the dishonor of dying in bed of amoebic dysentery) and of those close to the protagonist. The effect of the death on others can be reveal how the protagonist impacted their lives. Do they focus on the pointless death or the rich life? Do they fall to despair or find new courage as they try to honor the memory of the protagonist?
Finally, as has been pointed out in other answers, even if the reader has a strong favorable attachment to the protagonist, it is possible to transfer the mantle of significance to one or more other characters after the protagonist's death. This may be most easily done if there is a common cause championed by the protagonist such that the protagonist's death is largely a passing of the baton. However, if other characters are well-developed before the death, they may naturally fill the significance vacuum, especially if their initial actions are significantly informed by the protagonist's life and loss (not just taking up slack and mourning but also discovering how significant the protagonist was to them).