Text: No, because at the time of enrollment into a prospective cohort study nobody knows who will get the outcomes, and they are all part of the same cohort with common eligibility criteria, i.e., the entire cohort may be healthier than the general population. However, the comparison group is an internal one. It comes from the same healthy cohort. Consequently, if healthy people are more likely to volunteer for a cohort study, one might question whether the results are applicable to the broad population (external validity or generalizability), but at least for the study population the measure of association will not be biased the internal validity is ok.
In studies of screening programs, if one asks for volunteers to be screened, there may be a tendency to get health conscious volunteers. If outcomes in this group are compared to a general cohort representing the overall population (an external comparison group), then there will be a selection bias. This is similar to the healthy worker effect, except that here the screened population may have better outcomes (overestimate of benefit).