Rental car Drive from Prague to Rome? Good or Bad idea?

I will give the standard answer: Have you costed the charge for an international one way hire?
There is no problem visiting other countries with a hire car, but hiring a car in one country and dropping it off in another usually attracts a high surcharge.

You will also need to buy an Austrian Autobahn vignette, and pay Autostrada tolls in Italy.
http://www.viamichelin.com/ is a good site for getting routes, diostances and times, and it gives you fuel and toll costs. Praha to Roma:

Costs 229.23 € with a Petrol vehicle
Toll 57.40 € | Petrol 151.31 € | Road tax 20.52 €
Time 13h36 with 12h07 on motorways
Distance 1303 km with 1261 km on motorways

I wouldn't recommend it. In addition to the very good points Chris made, there is also a time consideration. North Americans are used to driving everywhere. There are times in Europe that a car is good to have. Mostly those involve exploring the countryside where public transportation is lacking. For example, exploring Tuscany and visiting the hill towns. It generally does not make time or financial sense to drive just to get from point A to point C, with a stopover at point B. Fuel is expensive. Driving can be slow. Why not take the train from Prague to Salzburg if that's your choice? Consider flying to Rome from there. If a stopover in Florence is your choice I would definitely fly, as it is a very long train trip. Take the train to Rome from Florence.

Dollar wise and time wise I couldn't think of a worse idea.

If you were a European with your own car, it would be one thing. But with a rental car it would be expensive because of the hundreds of Euro in international drop off fees. Also time wise those distances are best covered by air or high speed train.