My question is: how will the results of multi-core programs get affected if I have some CPU intensive applications running in the background? Will the quality of my results be affected by these applications?
Assuming the multi-core programs in question do not have bugs in them, such as race conditions, the results should be the same, although they will probably take longer to complete since some of the CPU cycles they would have used will be allocated instead to the other programs.
If the programs in question are time-sensitive (e.g. audio/video players, video games, robot-controllers, etc), the slower performance might in itself be a problem, since a time-sensitive program that doesn't get CPU time when it wants it can result in glitches in the playback, or a video game updating too slowly to be enjoyable, or the robot not responding quickly to keep itself from falling over, or etc.
But for programs that do strictly non-real-time batch/computational work, the computational results will be the same.