To save battery, I have created a "screen off" profile with SetCPU that locks the CPU frequency to the minimum while the screen is turned off. Could this have any negative side effects on apps that run in the background?
There will only be problems if you rely on background services to do processor intensive tasks. For example, on an older Android phone, music running in the background would skip when the CPU was set at a low enough clock rate.
So to answer your question, only if it's noticeable to you.