It seems likely that people want to switch from 2D to 3D when a simulation becomes turbulent.
At low Reynolds number the flow is laminar. In this case, it is reasonable to restrict your simulation to 2D if your problem has sufficient symmetry to allow it. A case in point is a flow around a cylinder in which the two walls at the end of the cylinder are connected by a periodicity condition.
As you increase the Reynolds number, the flow becomes turbulent. In this case, you can’t model 3D physics with a 2D numerical model, because the physics of turbulence is inherently different between the 2D and the 3D case. Even if your geometry has many symmetries, your flow wants to do things it can only do in 3D, such as, create 3D vortex structures, or transfer energy along different length scales according to the Kolmogorov theory.
There is no simple way of telling you at which point your simulation is turbulent (and thus, you need to switch from 2D to 3D). For well-known problems, just dig into the literature; it’s likely that other researchers have reported the value of the critical Reynolds number which marks the transition from laminar to turbulent flows. Otherwise, the rule of thumb is to say that as long as your simulation is stationary or time-periodic, it is laminar. If on the other hand you can’t observe any time periodicity, then you’re at risk of turbulence. But this is a test you need to do in 3D, because the critical Reynolds number is usually not the same in 2D as in 3D.