Replacing a 38c tire with 30c
The bike would sit about 4 mm lower is the main thing (and technically, the gearing will change slightly). 

You can mount the tire on there and it will work, but it may not be optimal depending on your rim width; if the rim is too wide for the tire, you may get more flats or rim damage when you hit a road hazard.

As for why you want to switch to something slimmer, if its for rolling resistance, you should read this and this.