Occasionally city-states will request that you build a road to connect them to your trade network, giving influence when you do. Is there any advantage to building a road to a city-state before they request it?
No. You can't form trade routes with city states, (at least with vanilla/G&K), so there isn't much a of a point to paying the cost of a road for nothing. Unless you want to set up a route for invasion... ;)
See Affine's comment if you're wondering about BNW
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If you are friends/ally of that particular city-state, then, from my experience, you get some bonuses related to that city-state's type. Example - Building a road to a cultural city-state will grant more culture per turn.