Why is it advised to avoid bubble formation during mixing?
A "bubble" per se would do no harm unless you have cells (without walls like mammalian cells) in the suspension. If you agitate the protein mixture vigourously then it may lead to denaturation of proteins by extensive intermolecular collisions. The "froth" formation is an indication of denaturation as denatured proteins stabilize these foams [1, 2]. 


References:


Zayas, Joseph F. "Foaming properties of proteins." Functionality of proteins in food. Springer Berlin Heidelberg, 1997. 260-309.
Schmidt, Isabelle, et al. "Foaming properties of protein/pectin electrostatic complexes and foam structure at nanoscale." Journal of colloid and interface science 345.2 (2010): 316-324.