As always, the choice is "it depends".
You're always giving up the bonus from the improvement when you settle on a tile. As mzprox and Trickster suggest, if the improvement provides +1G then it's a no-brainer to settle on it as long as doing so doesn't affect the other resources you enjoy from the city. You'll improve a different square with a Trading Post for +2G and come out ahead on the bargain.
It gets a little trickier when dealing with Food/Hammer improvements. For instance, a Gems tile goes from 0F/2H/4G to 2F/2H/4G when settled. But that same Gems tile becomes 0F/3H/5G if you improve it. +1H is a lot to give up; it comes out to more along the lines of +4/+5G. Here you have to balance the value of +3G on your city tile now against +1H and +1G once your city has grown.
Similarly, you can settle on Grassland Cows for a 3F city tile. But you would work that tile as your first tile anyway, and so you're not gaining anything by settling on it unless there's another amazing tile that you could be working as well. You are giving up +1H later on to settle on the Cows, which hurts (and is functionally -2H during Golden Ages).
I find that the majority of your early Gold comes from luxury resales, and that accelerating that process is unhelpful since it takes the AIs a number of turns to start raking in the cash. So I tend to be willing to settle on Calendar improvements, but not things that can be Mined or Pastured. YMMV.