The envisioned sixth-generation (6G) networks anticipate robust support for diverse applications, including massive machine-type communications, ultra-reliable low-latency communications, and enhanced mobile broadband. Intelligent Reflecting Surfaces (IRS) have emerged as a key technology capable of intelligently reconfiguring wireless propagation environments, thereby enhancing overall network performance. Traditional optimization techniques face limitations in meeting the stringent performance requirements of 6G networks due to the intricate and dynamic nature of the wireless environment. Consequently, Deep Learning (DL) techniques are employed within the IRS framework to optimize wireless system performance. This paper provides a comprehensive survey of the latest research in DL-aided IRS models, covering optimal beamforming, resource allocation control, channel estimation and prediction, signal detection, and system deployment. The focus is on presenting promising solutions within the constraints of different hardware configurations. The survey explores challenges, opportunities, and open research issues in DL-aided IRS, considering emerging technologies such as digital twins (DTs), computer vision (CV), blockchain, network function virtualization (NFC), integrated sensing and communication (ISAC), software-defined networking (SDN), mobile edge computing (MEC), unmanned aerial vehicles (UAVs), and non-orthogonal multiple access (NOMA). Practical design issues associated with these enabling technologies are also discussed, providing valuable insights into the current state and future directions of this evolving field.