The other characters' reactions can be quite diverse. Dumbledore (if he were still alive in the fanfic) might have some prior knowledge or plan related to Harry's 'death'. He could be guiding the other characters from behind the scenes. Some of the minor characters might be afraid that without Harry, the wizarding world has no hope. They could either try to flee or start to form their own resistance groups. And then there are those who might suspect foul play and start their own investigations into Harry's disappearance and supposed death.
When Harry is thought to be dead in fanfics, Neville could step up in a big way. He might initially be in shock like the others, but then his Gryffindor courage would shine through. He could start to inspire the other students at Hogwarts to fight against the dark forces. The Weasley family as a whole would be mourning, but also they would be a source of strength for each other. They might use their large network of friends and family to look for any signs of Harry being alive. And characters like Luna could have some unique insights. She might believe that Harry is in some other realm or state and try to convince the others to look beyond the obvious explanation of his death.
Hermione would probably be devastated at first. She might throw herself into researching ways to bring Harry back or to find out if he's truly dead. Ron would be angry and feel guilty, thinking they should have protected Harry better. Ginny, on the other hand, would be heartbroken but also determined to carry on Harry's legacy.