I just purchased a 1920 home that has all original horsehair plaster. Most is in okay shape but I have read so much about it causing asthma, etc...that I think I want to get rid of it. At the same time I read what a terrible mess it is to take down. Should I just drywall over it?

 

Plaster is inert, the only thing that could attribute asthma with plaster, is -IF- the plaster is cracking and falling apart badly, causing a lot of dust in the house.
As far as your project, DON'T... just don't mess with it. Firstly Horsehair Plaster is one of the finest, and most notoriously costly wall finishes in the western world. So if the wall is fine, leave it.
Secondly, trying to take down plaster just makes a massive mess with dust and debris that'll take forever to clean from a home.
Thirdly, drywall is a terrible, terrible product, as it will harbour mold, and it's installation produces a fine particulate dust that is harmful to asthmatics.

Hope this helps.

 

And the final point is that paint, wallpaper or whatever is applied over the plaster seals it and acts as a barrier between it and the room.