Systolic array-based deep neural networks (DNNs) accelerators have recently gained prominence for their low computational cost. However, their high energy consumption poses a bottleneck to their deployment in energy-constrained devices. To address this problem, approximate computing can be employed at the cost of some tolerable accuracy loss. However, such small accuracy variations may increase the sensitivity of DNNs towards undesired subtle disturbances, such as permanent faults. The impact of permanent faults in accurate DNNs has been thoroughly investigated in the literature. Conversely, the impact of permanent faults in approximate DNN accelerators (AxDNNs) is yet under-explored. The impact of such faults may vary with the fault bit positions, activation functions and approximation errors in AxDNN layers. Such dynamacity poses a considerable challenge to exploring the trade-off between their energy efficiency and fault resilience in AxDNNs. Towards this, we present an extensive layer-wise and bit-wise fault resilience and energy analysis of different AxDNNs, using the state-of-the-art Evoapprox8b signed multipliers. In particular, we vary the stuck-at 0, stuck-at 1 fault-bit positions, and activation functions to study their impact using the most widely used MNIST and Fashion-MNIST datasets. Our quantitative analysis shows that the permanent faults exacerbate the accuracy loss in AxDNNs when compared to the accurate DNN accelerators. For instance, a permanent fault in AxDNNs can lead up to 66% accuracy loss, whereas the same faulty bit can lead to only 9% accuracy loss in an accurate DNN accelerator. Our results demonstrate that the fault resilience in AxDNNs is orthogonal to the energy efficiency.