I am designing a flyer for the screen and the client insists the logo should be 71mm exactly. Does the resolution plays any role on the viewer, some may see it bigger, some smaller? Thank you.
Millimeters, inches, centimeters, picas.. all do not translate universally to pixel sizes.
Pixels are not a physical object, there's no measured size for a pixel.
Physical measurements only relate to printed materials, never anything on screen.
You need to ask your client if a pixel size is acceptable. Perhaps show him/her what 71mm would be at 72ppi (201px), 150ppi (419px), 300ppi (840px), because it will be different based upon the resolution of the image.
The size of any image presented on a computer screen will change with the resolution settings of that particular computer. There is no assurance that an image on your machine will be the same size as that image on my machine.
Additionally, the software being used to display the image will create variations in the displayed size. Web browsers can be adjusted to different zoom levels, image editors always have zoom capability, even PDF readers allow adjustable zoom.
Your client should provide specific parameters relevant to the above considerations to enable you to accomplish his objective.