Can a fluid approach the speed of light according to the equation of continuity?

That simplified form of continuity equation assumes that the fluid is incompressible. That is only a valid assumption at low Mach numbers. I think a typical “rule of thumb” is that a Mach number less than 0.3 is required for the assumption to hold. So for the continuity equation to hold in that form requires a speed which is much less than the speed of sound which in turn is much less than the speed of light. You cannot use the continuity equation to achieve supersonic flow, let alone relativistic flow.

Note, that is not to say that supersonic flow is impossible, but rather that it is not possible simply by application of that form of the continuity equation. You need a form that accounts for compressibility. Superluminal flow is fundamentally impossible as no massive particle can reach c with finite energy.

You can check your hypothesis by closing the mouth of a pipe (from which water is flowing out) to different degrees. You will find that the flow velocity increases initially as the outlet area is decreased, reaches a maximum and then falls off to zero as you proceed to completely close the outlet. This is because, when the outlet area is very small, viscous forces and surface tension effects become dominant and retard the flow velocity.

That is one version of the continuity equation for sure. But that is not the most general one. As a matter of fact, that condition can be deduced from the incompressibility of a fluid as in a limit of the Navier-Stokes equation:

$$\rho(\partial_t + v \cdot \nabla)v + \nabla p = 0,$$

where $$\rho$$ is the density, $$p$$ is the pressure and $$v$$ is the velocity field in the fluid, assumed to satisfy $$\nabla\cdot v = 0$$. But even if you take everything under consideration in this description, the formalism will allow for fluids to have a speed higher than the speed of light.

The reason for this is that even the Navier-Stokes equation is a limit of its relativistic counterpart, the conservation of the stress-energy tensor:

$$\nabla^\mu T_{\mu\nu} = 0.$$

And from this equation, it is easy to show that your fluid will not propagate faster than light under the assumption that your $$T_{\mu\nu}$$ is physical.