No, there is no such rule. Even if all values are positive, it all depends on the distribution. Additive shifts will change the relationship between the mean and the SD, but not change anything essential. E.g. take IQ. Right now, it's scaled so it has a mean of 100 and an SD of 15. But if you subtracted 50 from every score, the mean would be 50 and the SD would still be 15, but the amount of spread would be the same.
Beyond that, suppose we are measuring the heights of humans.
if we measure a sample of ALL humans (all ages) vs. adults (men and women) vs. men vs. basketball players, the mean will certainly change (each will be bigger than the one before), but the SD will change more (each will be smaller, I would bet).
If any values are negative, all bets are off.