There isn't a simple yes or no answer to your question. Your DVD writer needs to be fed with a steady stream of data to burn to the DVD. Ideally, the source of the data should be faster than the DVD burning rate so that the DVD writers buffer is always full or almost full. If you are running a program that makes intensive use of your CPU or uses a lot of disc I/O, you can easily get into a situation where the flow of data to the DVD writer falls below the burning rate. If this interruption is longer than a few hundred ms, the drive's buffer will run out of data and DVD burning will have to stop. This is called a 'buffer underrun". Back in the bad old days, a buffer underrun would result in a wasted disc or 'coaster'. Modern drives have buffer underrun protection so that they can recover from this situation.
Drives with buffer underrun protection can suspend the burning process and wait for the drive buffer to catch up. Once the buffer is full again, burning can resume as normal. A small number of underruns shouldn't cause too much degradation of the burn quality. A large number of buffer underruns is likely to cause severe problems.
It is best to avoid running CPU or I/O intensive programs while you are burning a DVD. Audio or video encoding (MP3/MPEG etc), action games, DVD playing or copying large files can lead to problems.
If you have a reasonably fast PC, you don't need to keep your hands in your pockets while the disc is burning, but you should avoid doing anything that is like to cause problems. Reading messages on CDFreaks, composing or reading e-mail and similar tasks shouldn't interfere with your burns.