We've replaced an old tub with a new one and put in a new drain/overflow kit using putty but have found a leak at the drain under the tub. Can we seal around the drain with silicone to stop the leak?

Nope. How did you make the seal?

Basically you create a 1/4" to 3/8 diameter length. Clean both surfaces. Put it under the lip of the drain. Put the rubber washer on the bottom, then a flat washer and squeeze the excess out.

If tightening the nut does not stop the leak then you must start over.