Casimir effect does create a region with negative energy density in the sense of general relativity, where such effect is said to violate energy conditions or create exotic matter. The topic is popular because such exotic matter can be used to stabilize traversable wormholes, which otherwise collapse without letting anything pass through, and then to form "time machines". While time travel enthusiasts are happy about it relativists are looking for an energy condition that holds even in the presence of quantum effects, but would guarantee that closed timelike curves can not form. See Visser's paper.
There is a big caveat. To be sure about this one needs a theory that combines quantum field theory and general relativity, and no such theory exists at present. Conclusions about negative energy density are based on semi-classical calculations, where classical metric tensor is coupled to quantum fields/particles. Such coupling leads to violations of fundamental principles of quantum field theory and/or general relativity, and semi-classical models are therefore self-inconsistent. From Rickles's paper (p.20):"...a classical field coupled to a quantized source will violate the uncertainty principle, since one will be able to use the classical field to determine with a precision greater than that allowed by the uncertainty relations the simultaneous position and momentum of a particle. Furthermore, if we adopt a collapse interpretation of quantum theory, so that the classical field’s measurement sends the particle’s state from a superposition into a definite state, then the principle of conservation of momentum is violated. If we adopt a no-collapse interpretation, then it becomes possible to exploit the coupling to transmit superluminal signals". Superluminal signals in relativity are also well known to enable time travel.
This does not necessarily mean that semi-classical gravity does not approximate the correct theory, Bohr's model of an atom was inconsistent but correctly derived the Rydberg formula for spectral lines. Moreover, the Hawking radiation was predicted by semi-classical gravity, and confirmed. But the inconsistency does manifest itself in the black hole information problem (see Rickles p.10).