If a trait had no effect on an organism's fitness what would happen to the allele for that trait

Perhaps nothing. Any trait that natural selection is blind to could pass, but many would pick-up deleterious mutations as the organisms possessing such traits would not be eliminated from the gene pool. Great ape vitamin C synthesizing pathways are one example of this. One gene coding for a part of this pathway has been ' knocked out ' by mutations and great apes (including us) can not synthesize vitamin C.