The main goal of the EXOZODI survey is to detect and characterize circumstellar dust and to propose the first statistical study of exozodiacal disks in the near-infrared using telescopes in both hemispheres (VLTI and CHARA). For this purpose, Ertel et al. have conducted in 2012 a survey of nearby main sequence stars with VLTI/PIONIER to search for the presence of circumstellar dust. This survey, carried out during 12 nights, comprises about 100 stars. For each star, we obtained typically three OBs and we searched for circumstellar emission based on the measurement of squared visibilities at short baselines. A drop in the measured visibilities with respect to the expected photospheric visibility indicates the presence of resolved emission around the target star. It is however generally not possible to conclude on the morphology of the detected emission based solely on the squared visibilities. Here, we focus on closure phases to search systematically for faint companions around the whole sample. Indeed, to derive robust statistics on the occurrence rate of bright exozodiacal disks, we need to discriminate between companions and disks. For this reason, the main goal of this paper is to discriminate between circumstellar disks (which show no closure phase provided that they are point-symmetric) and faint companions (point-like sources, creating non-zero closure phases). We also aim to reveal new companions that do not necessarily produce a significant signature in the squared visibilities, as the signature of the companion may show up more prominently in the closure phases. In this process, we reveal four new stellar companions with contrasts ranging from 2% to 95% (i.e., up to near-equal flux binaries). We also tentatively detect faint companions around one other target that will require follow-up observations to be confirmed or infirmed. We discuss the implications of these discoveries on the results of the exozodi survey.