  The case of SN Ia Hubble diagrams from photometrically selected samples using
photometric SN host galaxy redshifts is investigated. The host redshift
uncertainties and the contamination by core collapse SNe are addressed. As a
test, we use the 3-year photometric SN Ia sample of the SuperNova Legacy Survey
(SNLS), made of 437 objects between 0.1 and 1.05 in redshift. We combine this
sample with non-SNLS objects of the JLA spectroscopic sample, made of 501
objects mostly below 0.4 in redshift. We study two options for the origin of
the redshifts of the photometric sample, either provided entirely from the host
photometric redshift catalogue or a mixed origin where 75% of the sample can be
assigned spectroscopic redshifts. Using light curve simulations subject to the
same photometric selection as data, we study the impact of photometric redshift
uncertainties and contamination on flat $\Lambda CDM$ fits to Hubble diagrams
from such combined samples. We find that photometric redshifts and
contamination lead to biased cosmological parameters. The magnitude of the bias
is similar for both redshift options. This bias can be largely accounted for if
photometric redshift uncertainties and contamination are taken into account
when computing the SN magnitude bias correction due to selection effects. To
reduce the cosmological bias further, we explore two methods to propagate
redshift uncertainties into the cosmological likelihood computation, either by
refitting photometric redshifts with cosmology or by sampling the redshift
resolution function. Redshift refitting fails at correcting the cosmological
bias whatever the redshift option, while sampling slightly reduces it in both
cases. For actual data, we find compatible results with the JLA ones for mixed
photometric and spectroscopic redshifts, while the full photometric option is
biased but consistent with JLA when all uncertainties are included.
