Compare two hazard ratios from two subgroups? For example, for the smoker group, a cox regression shows that the hazard rate of alcohol use on survival is 1.41. In the non-smoker group, another cox regression shows that the hazard ratio of alcohol use on survival is 1.43. How do I know if 1.41 is significantly different from 1.43? In other words, does the smoking status affect the effect of alcohol use on survival?

Separate analyses on subgroups tend to lose power. The best way to handle this is to do a single Cox regression with smoking, alcohol, and their interaction included as predictors. If "smoking status affect[s] the effect of alcohol use on survival" significantly based on your data set, then the interaction coefficient estimate will be significantly different from 0 (hazard ratio significantly different from 1).
It seems unlikely that a hazard ratio of 1.41 will be statistically significantly different from one of 1.43. More important than statistical significance is the practical significance: would such a small difference actually matter in practice, even if it were "statistically significant"?
