Stay on the safe side - Explore at first the smallest size. Let the line thicknesses scale up for larger sizes. Of course that's not obligatory. Many fonts stay in good shape altough line thickness do not directly follow the tallness.
You must be sure, if there will be some texts that must be readable in the smallest size. If not, then find the smallest design elements that should have some visible details just like some text of the same size.
Then write a piece of random (=meaningless) text using font of the same size. Random, because a word can be recognized as whole in smaller size.The font should be "single stroke" (=no colored areas, only the curve)
Test, how thick font curve(* is readable at the proper distance for that logo size. That's your thinnest line for that logo size.
This needs some common sense and feel, because thinner line may be enough, if font is made taller. Its more about the total acreage (=area) of glyph's line than line thickness.
To be more safe do not use thin lines at all, only differently colored surfaces - in fonts too.
Check carefully, how the colors should be presented as greyshades. The logo must be well distinguishable also in BW.
*) When fonts are curved (=converted to outlines) you can explore the effect of the line thickness directly without making glyphs taller or smaller - only change the stroke.