Can I work out with sore muscles?

Senior research fellow in exercise physiology, John Saxton, replies: Delayed onset muscle soreness, or DOMS, is the sensation of pain and muscle stiffness which peaks two to three days after exercise, particularly if the workout involved the eccentric action of involved muscles. This is where active muscles are forcibly lengthened.
Many movements, including running, use a large component of eccentric muscular work and this type of muscle action is exaggerated during the controlled lowering of resistance weights. Any increase in the volume or intensity of eccentric muscle action during training will increase the likelihood of DOMS. The pain is associated with microscopic tears within the muscle fibres that generate a prolonged and localised inflammatory response.
Low to moderate levels of DOMS prior to the start of your workout is nothing to worry about and is related to the ongoing repair and regeneration processes in your muscle fibres. In fact, further exercise will temporarily alleviate any residual DOMS, but with the pain returning after exercise until the micro-injury has healed. There's no scientific evidence to show further exercise will exacerbate this level of micro-injury or be detrimental to health. However, muscles adapt fairly rapidly to this type of micro-injury, making them resistant to further DOMS if training intensity progresses gradually. So the regular occurrence of severe DOMS, in the absence of any notable change in training, may warrant medical investigation.