Why should glycerol not be added, to stabilize antibodies in serum?

Antibodies present in serum
Serum is a very stable format for antibody storage. In -20°C or -80°C, serum can usually be stored for years. In some specific cases, the shelf-life can be shorter for anti-peptide antibodies. For very short periods of time, serum may be stored at 4°C. In some cases, more careful freezing with a first step at -20°C, followed by -70°C is beneficial. Snap freezing in liquid nitrogen is not required. 

Total IgG fraction (IgG antibodies purified on Protein G matrix)
Generally, protein G purified antibodies are stable. They can be stored in -20°C or -80°C for years. For short-term storage, add a preservative like 0.2 % ProClin300 (Sigma), which is less toxic alternative, compared to sodium azide.

Total IgY fraction (IgY antibodies purified by precipitation from egg yolk)
Purified IgY fractions are very stable, even at room temperature (although we do not recommend it as storage conditions). IgY can be stored at +4°C with 0.02 % sodium azide (note: azide inhibits activity of HRP enzyme), gentamicin sulfate (50 µg/ml). Avoid freezing and thawing of IgY, and storing it on dry ice. IgY antibodies can be stored at -20°C.