Can I damage a biotinylated antibody in solution by vortexing or mixing too vigorously?
It is good practice to avoid vortex mixing any protein solutions, especially vigorously.  Vigorous vortex mixing can produce foaming and proteins can denature (unfold and lose activity) when they are present at an air/liquid interface (as in a foam).  This is because the hydrophobic regions of the protein will want to be in the air region and the hydrophilic regions will want to be in solution, and so unfolding may occur.
In practice, antibodies are usually quite stable as proteins go and you can often vortex them without much negative effect to your experiment. That said, if you routinely vortex proteins in the lab and come across a less stable one you will suffer degradation.  Since you will not always know in advance which proteins will be OK to vortex, it is best practice to mix gently using a pipetman (again avoid introducing air that creates a foam) or inversion, or other method. If you must vortex, then be gentle with short pulses, avoiding foaming.
I have purchased some antibodies where the datasheet specifically says not to vortex mix.