Yes it does. It's enough to e.g. train a slave, give a travel command and assign a worker. You can feel (and see, if you look on a clock) that the time required to save goes up. Even if it is only about 2-3 seconds after something like this.
Even if I do nothing there is still a small delay, about a second or so, that wasn't there when I started with this playthrough. But I thought that might have something to do with the time I already played. I'm in week 10 now and have 12 characters in my household, guesing the complexity of the savegame information might have increased or something changed as the game was updated. But I'm fishing in the dark here.