There are three corniches -- the upper one is the worst (Grande Corniche of Princess Grace fame).
The first path in the Cinque Terre is wide and paved. By about the third/fourth, things get higher and narrower and you'll not want to go as it turns into a goat scramble.
Eze is pretty high, with a large drop off, if your fiance does not like high overlooks. Definitely don't take the path down. The next town up, La Turbie, has some steep dropoffs, especially from the Trophee des Alpes. There are also some twisty narrow road with dropoffs as as you head into the pre-Alpes, but most people don't go on their first trip (or they take a train).
The path up to Castle Hill in Nice is wide and paved (a staircase -- you can take an elevator), but the overlook does have some height to it (a great view, but if they are especially sensitive). Same thing with the view down from le Rocher in Monaco. The edge of Gourdon, from the village down to the Gorge du Loup is also a large drop. If they are fine with staying back from the edge or observing from a distance, there is no reason to avoid any of these.