If Luke and Leia were not siblings, then the entire plot would begin to unravel.
It was only dumb luck that Han Solo came along for the ride in the first place. He was simply a mercenary when he met with Luke and Obi-Wan for the first time. All he genuinely cared about was getting enough money to pay off Jabba the Hutt because there was a bounty on his head.
The only reason why the Han character was introduced in the first place, was to transport Luke and Obi-Wan to Alderaan. In theory, he would have been paid for his services and continued on his way. In fact, Alderaan had already been destroyed by the Death Star. This is when the plots of Luke and Han (and Obi-Wan, Leia, Chewbacca, R2-D2, and C-3PO) became entangled.
Without Han's help, then it would be unlikely that Luke and Obi-Wan would be able to rescue Leia alone. She was essentially the glue that bonded all of the characters together. If it wasn't for her being abducted (and sending R2-D2 and C-3PO for help to Obi-Wan), then none of these characters would have met.
Lets look back to the first scene when Leia recorded a message for Obi-Wan. She knew where he was, but didn't have any clue about the existence of Luke. The mission of C-3PO and R2-D2 was to find Obi-Wan, not Luke. It was only dumb luck, or perhaps The Force that brought them together. If the droids found Obi-Wan first, then it would be awkward for Obi-Wan to recruit Luke. Luke was only really compelled to leave in the first place because his family was slaughtered, and he had nothing left to stay there for. If the droids never made it to Uncle Owen and Aunt Beru's farm, then they would have been unharmed.
After the Leia was rescued, by Han and Luke, then a love triangle was formed. The only way to civilly resolve the issue was to make Luke be a sibling to Leia. This easily and definitively resolved the love triangle. If Leia did not have a love interest in Han, then there would be little need to keep his character going after the first film.
What kept Han Solo going through the whole trilogy was the mutual love interest between him and Leia. This connection eventually led to him joining and aiding the rebellion. Without his help, then the mission to destroy the first Death Star would have failed. Luke would have died, and the rebellion would have been ended.