Personally I found that it is better to take some time off working out when you feel sick. Don't forget that your body needs rest too and the sickness may be your body trying to tell you to take it easy for a while. Also, while you are recovering you can still use the time to improve other parts of your life. Your physical fitness is important, however, there is more to self-improvement and hapiness than just working out.
I think you should always listen to your body.if you feel really sick and tired,then don’t exercise, take a one or two day rest. You might feel better and rested you can then start doing your usual exercise routine and trust me you will not lose your progress. Personally this is what works for me!!
Sickness is usually a sign that you've overexerted your body a bit so it's better to rest for atleast a little bit or make your exercise less intense so your body can heal and rest. Instead of exercising you can also just do a good stretch to keep your body mobile and also try not to skimp on the rest of your regular routines while being sick like eating healthy and all your regular dailies. Skimping on those while you're sick makes it really hard to get back into the habit of exercise especially if they're related to your body.
When you have a cold or a sickness you should definitely not run if I am sick my top priority is getting healthy again so if I feel like I can do it yes but I do not push myself if I do not feel like doing it.