he would have nipped the tetra to death or near death, the other fish (especially the tiger barbs go for crippled or sick fish) would then finish the job and then most likely eaten the carcase.
tiger barbs and red-eye tetra's are known to get along very well. how many tiger barbs and how many red-eye tetra's do you have in the tank? it could be that there numbers were not large enough to spread the aggression of the red-finned shark.
people say you need 5 tiger barbs or 5 red-eye tetra's to complete the school and spread aggression.. i feel that one should not go with less than 8 or nine of either of these for a proper school and to control the agression.
i think your tank mates are compatible, its probably just that you dont have enough of each type of fish (besides the red-finned shark - you only ever keep one of those in a tank)