Elaborating on a comment I made earlier: we perspire, and some of that perspiration will hit your bike or your handlebars. As stated elsewhere, you want to wipe your bike off, and you should consider some sort of covering for the top tube. Admittedly, this is not the trainer itself damaging your bicycle. However, outdoors, you're moving through air, which dissipates a lot of heat. I'm not sure this has been scientifically studied, but received wisdom appears to be that indoor cooling is almost always less effective than the cooling you get from riding outdoors, unless you invest in many powerful fans. Thus, dripping perspiration is an inevitable consequence of indoor training.
If you are using a steel frame, it should be obvious that this is bad.
If you have a carbon frame, you will have aluminum fittings for things like the bottle cage bosses. Perspiration can facilitate galvanic corrosion over time. The image below is reproduced from a Facebook post by Raoul Luescher, an Australia-based carbon repair expert. He described the white powdery stuff as the aluminum basically disintegrating due to corrosion.
It may also be possible for aluminum frames or components to corrode from sweat in some circumstances. This Roadbikereview forum post contains three pictures, two of which are reproduced below. The rider said that they had gone through two of this model of handlebar in three years.
This article by Wiley Metal says that while raw aluminum normally has a protective oxide layer forming immediately, sodium chloride or other chloride salts can attack that oxide layer. To protect against that, we normally paint or anodize aluminum components, and all aluminum handlebars are indeed anodized. This increases the depth of the oxide layer and I believe it also makes it porous enough to accept dyes. Most handlebars are dyed black, and I think they're also covered with a clear coat. However, when mounting your shifters on your bars, you may introduce surface scratches that expose the oxide layer. The poster above may not be typical, e.g. they might perspire a lot more and a lot saltier* than average. However, for this reason, it's worth changing your bar tape every so often, just so you can check the status of the handlebars, and perhaps clean off accumulated salt deposits if there are any.
*While not directly related to the original question, you do need to replenish electrolytes during endurance activity. Individuals do vary in how salty their perspiration is. If you notice yourself craving salty foods after or during a ride, you should strongly consider getting some electrolyte supplements if you don't already use them. They needn't be used on every ride, but the benefit can be substantial on longer rides, especially in the heat. If you figure that your sweat is saltier than your peers, it might also be worth checking up on your handlebars even if you don't do indoor cycling.