Gamma-ray flares of blazars may be accompanied by high-energy neutrinos due to interactions of high-energy cosmic rays in the jet with photons, as suggested by the detection of the high-energy neutrino IceCube-170922A during a major gamma-ray flare from blazar TXS 0506+056 at the ~3{sigma} significance level. In this work, we present a statistical study of gamma-ray emission from blazars to constrain the contribution of gamma-ray flares to their neutrino output. We construct weekly binned light curves for 145 gamma-ray bright blazars in the Fermi Large Area Telescope Monitored Source List adding TXS 0506+056. We derive the fraction of time spent in the flaring state (flare duty cycle) and the fraction of energy released during each flare from the light curves with a Bayesian blocks algorithm. We find that blazars with lower flare duty cycles and energy fractions are more numerous among our sample. We identify a significant difference in flare duty cycles between blazar subclasses at a significance level of 5%. Then using a general scaling relation for the neutrino and gamma-ray luminosities, L_{nu}_{propto}(L_{gamma}_)^{gamma}^ with a weighting exponent of {gamma}=1.0-2.0, normalized to the quiescent gamma-ray or X-ray flux of each blazar, we evaluate the neutrino energy flux of each gamma-ray flare. The gamma-ray flare distribution indicates that blazar neutrino emission may be dominated by flares for {gamma}>~1.5. The neutrino energy fluxes for 1 week and 10 yr bins are compared with the decl.-dependent IceCube sensitivity to constrain the standard neutrino emission models for gamma-ray flares. Finally, we present the upper-limit contribution of blazar gamma-ray flares to the isotropic diffuse neutrino flux.