Many future Internet of Things (IoT) applications are expected to rely heavily on reconfigurable intelligent surface (RIS)-aided unmanned aerial vehicles (UAVs). However, the endurance of such systems is constrained by the limited onboard energy, where frequent recharging or battery replacements are required. This consequently disrupts continuous operation and may be impractical in disaster scenarios. To address this challenge, we explore a dual energy harvesting (EH) framework that integrates time-switching (TS), power-splitting (PS), and element-splitting (ES) EH protocols for radio frequency energy, along with solar energy as a renewable source. First, we present the proposed system architecture and EH operating protocols, introducing the proposed hybrid ES-TS-PS EH strategy to extend UAV-mounted RIS endurance. Next, we outline key application scenarios and the associated design challenges. After that, a deep reinforcement learning-based framework is introduced to maximize the EH efficiency by jointly optimizing UAV trajectory, RIS phase shifts, and EH strategies. The framework considers dual EH, hardware impairments, and channel state information imperfections to reflect real-world deployment conditions. The optimization problem is formulated as a Markov decision process and solved using an enhanced deep deterministic policy gradient algorithm, incorporating clipped double Q-learning and softmax-based Q-value estimation for improved stability and efficiency. The results demonstrate significant performance gains compared to the considered baseline approaches. Finally, possible challenges and open research directions are presented, highlighting the transformative potential of energy-efficient UAV-mounted RIS networks for IoT systems.