There are always incredibly frustrating ways experiments can go wrong, even at a professional level. Were the two solutions well-mixed? If you are mixing two solutions that start at different pHs, the resulting solution will always be somewhere between the two initial pHs. If you are seeing no change, it is possible one of the solutions (either KIO₃ or orange juice) was not at the initial pH you suspected. If the pHs are too close, there won't be much change in the final solution. If the pHs are significantly different but you see no change, perhaps you did not add enough of the base to raise the orange juice pH. This would be fixed just by adding more KIO₃.

As always (and especially in high school labs) the chemicals could be very impure or contaminated, so that could cause issues.

If you add significant amounts of KIO₃, the pHs are very different before mixing, and you STILL see no change, the solution could be what's called a "buffer." (If you already know what a buffer is, feel free to skip the rest of this answer.) Buffers are solutions that are relatively stable in pH despite addition of acid or base. This is done by weak acids or bases exchanging free hydrogens with the strong acid or base, preventing too much change in the ion content of the solution. Most chem textbooks discuss this more thoroughly, and this link is a good resource. Many biological fluids (the textbook example is blood) are buffers because if you didn't have molecules stabilizing pH, the fluids would be very sensitive pH changes in the environment and would likely kill the organism. I don't know enough to say for absolute certain that this is a buffer reaction, but it may be.

Ascorbic acid is a fairly weak acid, and potassium iodate may be chemically active itself. Maybe the concentration of ascorbic acid was so large that you couldn't titrate it all, or that there were too many other things (e.g. citric acid) in the orange juice for it to matter.