  We study the effects of gravitational lensing by galaxy clusters of the
background of dusty star-forming galaxies (DSFGs) and the Cosmic Microwave
Background (CMB), and examine the implications for Sunyaev-Zel'dovich-based
(SZ) galaxy cluster surveys. At the locations of galaxy clusters, gravitational
lensing modifies the probability distribution of the background flux of the
DSFGs as well as the CMB. We find that, in the case of a single-frequency 150
GHz survey, lensing of DSFGs leads to both a slight increase (~10%) in detected
cluster number counts (due to a ~ 50% increase in the variance of the DSFG
background, and hence an increased Eddington bias), as well as to a rare
(occurring in ~2% of clusters) "filling-in" of SZ cluster signals by bright
strongly lensed background sources. Lensing of the CMB leads to a ~55%
reduction in CMB power at the location of massive galaxy clusters in a
spatially-matched single-frequency filter, leading to a net decrease in
detected cluster number counts. We find that the increase in DSFG power and
decrease in CMB power due to lensing at cluster locations largely cancel, such
that the net effect on cluster number counts for current SZ surveys is
sub-dominant to Poisson errors.
