When children are ill, the type of illness would dictate the level of regression.
When sick, you should not go back to being the prop (putting your child to sleep), but you can be there for your children in other ways so that they are not suffering alone.
Teething overall can be an excuse as to why children are up through the night. When babies/toddlers have good sleep habits prior to teething, wake-ups due to teething would be minimal compared to a child who had poor sleeping habits to begin with.
With teething, you also find that some teeth can come in without you even knowing and some are more bothersome.
The bottom line is that if through the night your child needs you, then you should make them comfortable. Regression usually seeps in when you go back to adding props in an attempt to put your child to sleep during these phases.
If you do go off track, then once you are back in a place where all is well, you may have to do some retraining. You should not do this though until you are confident that your child has fully recovered.