How long does it take unused antibodies to clear the system?
I agree that this question may be broader than a simple half-life question in theory, however, I think that it is still basically a half-life question. For instance, does the presence of the virus (using the flu example) contribute to opsionization and clearance? Possibly, but this would be variable and difficult to generalize about. 

The question has been asked in a number of ways (for endogenous / exogenous antibody). For simplicity, Mankarious et al examined transferred antibody in IVIG therapy and found, "For total IgG, the half-life found was 25.8 days; for IgG1 it was 29.7 days; for IgG2 it was 26.9 days; and for IgG3 it was 15.7 days." - an answer in line with that proposed by canadianer, above.