Since employment status and income are self-reported, there is always the possibility of error. Looking across March CPS samples, the percentage of employed persons reporting zero income never appears to be more than 4% in any given year, and this percentage has been below 2% since 1980. Therefore, simply excluding these persons likely will not do much to harm the integrity of your sample. On the other hand, you will be disproportionately dropping young respondents. For example, while only 10% of employed persons reporting non-zero income were between the ages of 15-23 in 2009, 50% of employed persons with a zero income fell within this age range.
As an alternative to dropping all of these cases, you can likely salvage a sizeable percentage by making some reasonable assumptions (based on your judgement of what is reasonable). While the vast majority do identify as wage/salary workers, according to CLASSWKR, over 20% of these cases identify as self-employed or an unpaid family worker. It is plausible that these persons consider themselves as working without making any income. Using OCC and looking at 2009 again as an example, half of the private wage/salary workers who report zero income work in service occupations such as cashiers, cooks, waitresses, child care workers, counter attendants, laborers and home health aides. Again, one could plausibly see someone working at one of these occupations without reporting any income, e.g. working as a cook/waitress in a family business or regularly babysitting for a family member. In light of the extremely young age of this group, some of these cases could be something such as an unpaid internship or a work study program where the respondent is receiving pay but may not consider the earnings as income since it likely goes toward tuition/school expenses.
Unfortunately, there is ultimately no way to tell whether the zero income is an error or an accurate figure; therefore, the final call on how to handle cases of employed persons reporting zero income is up to the researcher’s discretion.
Hope this helps.