I need to do viscosity measurements at different temperatures in order to fit a model of the form (Flucher equation): $$\log_{10}{\eta}=A+\frac{B}{T-T_0}$$
Where $T_0$ is constant and known, $T$ is the temperature at which I'm making the measurement, $\eta$ is me the measured quantity, and $A$ and $B$ are the constants I'm looking for.
How to know the minimum amount of measurements I need to do to fit the model?
First small remark : Fulcher and not Flucher (to be more precise, this is Vogel-Fulcher-Tammann correlation).
As said in comments, since you have two parameters $A,B$, two points would be sufficient.
However, you precised that this is for viscosity and the problem is that the measurements of this specific property have never been very accurate. On the other hand, viscosity is very important in many areas such as design or rating of heat exchangers, pipelines , trays in distillation columns,and so on.
So, my answer would be : as many measurements as you can afford.
On the other hand, you must remember that, typically, $T_0$ is not known; it is one parameter to be adjusted.
My last point would be that this model extrapolates very poorly. So, try to really cover the range of temperature you will be concerned by.