3D concrete printing (3DCP) brings automation in construction, reduces material usage, increases design flexibility, and eliminates the need for formwork. However, it is a complex process involving various parameters that are often defined by trial and error. This can lead to unforeseen failures during the print, such as buckling or yielding. Computational modeling can be used in the design stage to predict and prevent failure, during printing for real-time process control, and afterwards to assess how variations during printing affect the final structure. The structural failure during the print is primarily governed by how concrete behaves at the material level, making the choice of constitutive model crucial. Plasticity models are commonly used to assess buildability, with the Mohr-Coulomb criterion being a widely used approach. However, its suitability for modeling fresh concrete for 3DCP, under such loading conditions and varying material properties is still an open research question. Furthermore, experimental studies have shown that fresh concrete exhibits non-linear behavior before failure, which is usually not considered in structural simulations of 3DCP. This work investigates the influence of plasticity models on different structural failure modes observed in 3DCP, specifically elastic buckling and plastic collapse. The non-linear behavior of fresh concrete is accounted for by incorporating non-linear isotropic hardening into the plasticity models. A Von-Mises plasticity model and a Mohr-Coulomb model with a hyperbolic smooth approximation are implemented, both incorporating non-linear hardening. An objective stress rate formulation is adopted to consider geometric non-linearity for accurate buckling predictions. As freshly deposited layers structurate over time, an age-dependent model is implemented to capture the stiffness and strength evolution of printed layers. To simulate the layer-by-layer process, a pseudo-density-based activation method is used, allowing sequential activation of layers as printing progresses. Model parameters are identified for different ages using Bayesian inference via inverse finite element modeling by numerically replicating stress-strain data from uniaxial compression tests on samples at different ages. Printing simulations are conducted for thin-walled and cylindrical structures, demonstrating the influence of choice of plasticity model on buckling behavior and material failure.