We are installing new pop up drains (the kind you push down on to close and open) on new sinks without overflow. The drain part sticks up about an eighth of an inch from the sink so after it drains there is always a little pool of water around the drain. The drain came with two rubber washers, one of which we already got rid of (it, too, caused the drain to be too high) and the other which is mounted under the sink. We used plumber's putty to create a seal between the drain and the sink. How much plumber's putty is needed to create a good seal? Is this what's causing the drain to stand above the sink hole? It isn't leaking, BTW. We have the same problem on 3 Kohler sinks and a vessel sink, all the same drains. Any other causes or ideas?

You use only one washer and that is on the nut under the sink. The plumbers putty should be rolled into a tube the thickness of a cigarette and then wrapped around the new drain immediately under its lip. One wrap only and pinch off any excess. When you tihten the nut under the sink the plutty sqeezes out and leaves nearly no raised height above the sink, then new drain seats almost recessed. I think maybe you used too much putty and did not tighten enough.

If the drain mechanism that goes through the sink is made of plastic that could be your problem. To make them have more strength they have to be a little thicker than if it were a brass piece, plus then you have the chrome ring added to it making it even thicker.
Another problem could be that if you sink is cast out of plastic or even if it is marble the hole that the drain flange should fit into is not deep enough and the drain can not seat the way it should to not have the water standing.