Agree with the advice to use a towel and clean off the bike when done if any sweat drips on the tubes. I use my 30 year old steel bike on the trainer, and have some corrosion on the underside of the top tube as a result of not keeping it clean.
There are special bike-protector towels which are made with elastic loops to go over the handlebars and seatpost, suspending the terrycloth above the top tube. Haven't tried them yet though.
On the headset, I don't see why any damage would be done at all with just your normal upper body weight resting on the bars. Grooving of the headset should only occur from high-impacts of slamming into and over road bumps, not bodyweight.
Dan