It is possible to install an A-frame roof over a flat roof without removing the flat roof. In fact, the procedure is common enough that some cities address it in their building permit application process. Some companies use this as their basic procedure when adding a metal roof to a home, installing beams and rafters into the existing decking.
Cities may have extra requirements for roof installation when the new roof is merely being placed over the old one. For example, the city of Lodi, California, requires plans to show how any air conditioning units left on the original roofing will be affected by the new roof. The city requires certain clearances and drainage access before it will issue a building permit.
Talk to your insurance agent about how your homeowner rates will change. You are reducing the risk of ceiling leaks, but you’re also increasing the risk of a fire. Deer Park Roofing warns that as the slope of the roof increases -- as it becomes steeper -- the fire risk increases because a fire will spread more quickly. If flames appear at the base of a pitched roof, the angle of the roof means more of the body of the flames will be closer to the roof. This is the same dynamic you find when a brushfire spreads to a hillside -- if the flames head up from the bottom, they travel more quickly.
Ensure that the roofing contractors you work with have experience installing pitched roofs over existing flat roofs. The contractors must be able to install all the necessary supports without damaging your ceiling inside, and must follow all local laws so that you don’t run afoul of any housing codes. Have any water damage fixed before installing the new roof to prevent damaged areas from being blocked off by construction or by support beams. This will also help preserve the integrity of the old roof as it’s now part of the support system for the new roof. If the old roof has any rotten spots, it could create a mold or rot problem requiring additional roof work after the new roof is installed.