Flushing a live fish down the toilet will not kill it but the fish will eventually die once the fish gets into the untreated sewage in the main sewer pipes or when it reaches the sewer treatment plant.
Fish can survive the flushing down the toilet part but the fish cannot survive in the actual sewage water when it reaches the sewage.
So if you flush a live fish down the toilet it will die eventually but not right when you flush it.
If you have any fish you want to get rid of you should place them in a river or lake if they are alive and not in a toilet.
But if the fish dies then disposing of the dead fish in the toilet would be okay.