If you demolish your academy, you will lose the researched skills. Under lvl 30 you can't organize city festivals, olympyc games or theater shows. Under lvl 28 you will lose the conquer ability (also the ram and cartography)...
Take care with your city wall if you have tower. The tower doesnt works with a wall under lvl 20.
A couple of errors in this statement.
1. Once a skill is researched in the academy, you own it. You WILL NOT lose the ability if you demolish your academy below the level that was required to research it.
2.
You DO NOT need an academy level 30 to run Olympic Games or Theatre plays (only gold or a theatre). (edit-you DO need level 30 for Games and Plays) You DO need an academy level 30 to run City Festivals - these can be important to increasing your Culture Points (depending on your style of play) which is why most people build the academy to 30 and keep it there.
3. You CAN demolish your wall as much as you want and there will be NO EFFECT on the Tower. Wall level 20 is only required to build the tower. Once a building (or building level) is built you acquire it's attributes and can demolish any requirements for that build without losing those attributes.
Besides, you can't demolish the Academy a level unless you have at least 4 unused research points.
This too is incorrect - you can demolish an Academy level any time you wish even if it would take you into negative research points. However, if you have built the Library, you must have 12 free research point to demolish it.