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Intrinsic alignment (IA) modelling and photometric redshift estimation are two of the main sources of systematic uncertainty in weak lensing surveys. We investigate the impact of redshift errors and their interplay with different IA models. Generally, errors on the mean $δ_z$ and on the width $σ_z$ of the redshift bins can both lead to biases in cosmological constraints. We find that such biases can, however, only be partially resolved by marginalizing over $δ_z$ and $σ_z$. For Stage-III surveys, $δ_z$ and $σ_z$ cannot be well constrained due to limited statistics. The resulting biases are thus sensitive to prior volume effects. For Stage-IV surveys, we observe that marginalizing over the redshift parameters has an impact and reduces the bias. We derive requirements on the uncertainty of $σ_z$ and $δ_z$ for both Stage-III and Stage-IV surveys. We assume that the redshift systematic errors on $S_8$ should be less than half of the statistical errors, and the median bias should be smaller than $0.25σ$. We find that the uncertainty on $δ_z$ has to be $\lesssim0.025$ for the NLA IA model with a Stage-III survey. For $σ_z$, the requirement is met even for large uncertainties $\leq0.3$. For the TATT IA model, the uncertainty on $δ_z$ has to be $\lesssim0.02$ and the uncertainty on $σ_z$ has to be $\lesssim0.2$. For Stage-IV surveys, the uncertainty on $δ_z$ has to be $\lesssim0.005$ and the uncertainty on $σ_z$ should be $\lesssim0.1$, with no significant dependence on the IA model. This required high precision will be a challenge for the redshift calibration of these future surveys. Finally, we investigate whether the interplay between redshift systematics and IA modelling can explain the $S_8$-tension between cosmic shear results and CMB measurements. We find that this is unlikely to explain the current $S_8$-tension.