This is a hard to answer question because a truly correct answer would involve static analysis of a given Intelligence to determine whether it has the computational capability to generate a looping state (e.g. some state which reproduces itself in the next instance), and in fact whether these looping states can even exist in the given architecture.
Generally speaking, assuming we can build AI, there is no reason it would be impossible build an AI that has the potential to get stuck in a computational loop and become non-functional. You can imagine a recurrent neural network where the continuous state inside the ANN dominates its behavior causing it to reproduce states with small variety (as input will always be relevant in a connected ANN) but are converged around a particular state space. Whether that is akin to death is a philosophical question for your own peace of mind.
However, if we designed an AI agent that commonly did this, that would reflect poor design; that wouldn't necessarily devalue the contribution of the designed agent but it would bring the agent's other activities under scrutiny. If this is a bi-product of some simulated intelligence, is the process going on really that intelligent?