Heat. A resistor dissipates power as heat. I don't know why a resistor is between I, which is 12 volts from the Ignition, and S, which is the AC signal from the stator, rectifier or power pack if you have one. Regardless of the source for S, it does comes from the stator, either from a rectifier or power pack connection.
Let's stop guessing. Hook up a meter on the AC scale to the S stud and the other lead to the ground stud, the one with the metal strip. Start the motor and see what AC voltage is shown and does it start high and drop off?
And do the earlier DC measurement across the battery. What are you seeing there?
One other thing - Did you replace the tach with the same brand and model? Did you transfer the resistor to the new tach? Did the new tach come with any instructions about a resistor?