Survival analysis is often done under the assumption of non-informative censoring, e.g. censoring is independent of failure time. To give an example of when this breaks down is not too difficult: think of the situation where censoring is clearly informative. An extreme example of this would be to censor every patient right before death, leaving us with only survivors.
The above example is of course silly, but a more realistic one is the case where patients undergoing different treatments have to report to some hospital every month. If not, they are censored from the study. Then censoring might become more likely when a patient's situation is getting worse as the patient doesn't have the energy to go to the hospital for some purely scientific purpose.
In your example of a wonder-drug, we can (under appropriate assumptions) still estimate survival using standard methods. If the drug takes some time to kick-in there'll be an increased hazard (compared to later) right after taking the drug, if we start follow-up right when the drug is taken. This seems only fair. But the drug will be assessed using all the patients in the study, also the censored ones, as they contribute survival time to the calculations.
Otherwise, you could include in your study only patients that had survived e.g. the first week after having received the drug. But then you'd be conditioning on surviving some period of time, of course.
Another quite common problem is the introduction of survival bias in terms of "immortal time". For instance, in some observational studies, the subjects may be known from some date, but only because they survived. Thus, the first immortal time (where had the subject died it would not have been included in the study) should be discarded when analysing data. A different example of immortal time bias can be studied by reading more on the Stanford heart transplant data. Originally, researchers working on this study calculated survival time for the patient given a heart transplant as the time from being accepted for transplantation until death or censoring. However, in the time from being accepted until the actual operation the patients were not exposed to the treatment (transplantation). This time should have counted as survival of unexposed patients, not exposed ones, as the researchers did.
In conclusion, bias is all around, but I don't see it in your example.