Should I continue to workout when I am sore?

If it is just stiffness and a little general soreness, it will help to work out as @noelleptc stated. If you are having localized soreness that intensifies as you workout, or a sharp pain you may have an injury that requires attention.

Sure you can, as @noelleptc said “It does help with the soreness.’

You don’t say what activities you are doing. If you are weight training than you work opposing muscle groups so you rest the others. Working opposing groups allows you to tax the muscles which causes little tears in the muscles which builds strength, tone and mass (mass if you work for that) and then rest allows them to recover thus preventing injury and aiding your increase in strength. If you are walking or jogging than maybe you are attempting to do too much to soon. Do less over all time and during your workout do several bursts of more intense activity and across the days increase your time and change up your intensities. You may also want to switch up activities. You could swim one day and do some spinning on another day. But you can continue to work out so long as you are not in real pain or have sharp pain as WestRiverrat mentioned.

you may want to back off some if you have a strain, sprain or some other mechanical injury that will be made worse. Sore muscles don’t count as much as reason to stop, I don’t think.

No. Listen to your body.