Machine learning (ML) models are widely used in many important domains. For efficiently processing these computational- and memory-intensive applications, tensors of these over-parameterized models are compressed by leveraging sparsity, size reduction, and quantization of tensors. Unstructured sparsity and tensors with varying dimensions yield irregular computation, communication, and memory access patterns; processing them on hardware accelerators in a conventional manner does not inherently leverage acceleration opportunities. This paper provides a comprehensive survey on the efficient execution of sparse and irregular tensor computations of ML models on hardware accelerators. In particular, it discusses enhancement modules in the architecture design and the software support; categorizes different hardware designs and acceleration techniques and analyzes them in terms of hardware and execution costs; analyzes achievable accelerations for recent DNNs; highlights further opportunities in terms of hardware/software/model co-design optimizations (inter/intra-module). The takeaways from this paper include: understanding the key challenges in accelerating sparse, irregular-shaped, and quantized tensors; understanding enhancements in accelerator systems for supporting their efficient computations; analyzing trade-offs in opting for a specific design choice for encoding, storing, extracting, communicating, computing, and load-balancing the non-zeros; understanding how structured sparsity can improve storage efficiency and balance computations; understanding how to compile and map models with sparse tensors on the accelerators; understanding recent design trends for efficient accelerations and further opportunities.