I'm not sure if you've conducted the survey already or if this is a hypothetical, but you should really decide on protocol ahead of time. As for what to do, you could totally exclude incomplete surveys. You could also do two separate analyses for the data, one which integrates all of the data and one which excludes partially completed surveys.
It depends on how many of the unfinished questionnaires you want to remove from the survey data? Practically speaking, if these are less in numbers, you can recheck your formula through which you had drawn the sample. You can edit the value of sample error and recalculate the sample size. I hope the unfinished questionnaires could be adjusted.
Depending upon how you attempt to infer to your population, you might have serious bias anyway, but the main problem with nonresponse is often bias. If your sample size is relatively small, variance may be a problem as well. (I worked with quantitative surveys, but I can see that there would be an analogous impact, though harder to 'measure,' for qualitative surveys.)
So if you have "item nonresponse," and throw out the whole survey, then you generally give yourself a bias problem for all "items" (questions/variables of interest/dependent variables), rather than just the items that were missing.
INCOMPLETE QUESTIONNAIRE should be treated as defective survey. This is called defect by incomplete response. if the removal will affect your minimum sample size requirement, for each removal you need to replace with a new one if it is feasible.
IF STILL EARLY of the research and if the non-responses are many, double check you questions in the survey to evaluate whether the non-responses came from the sensitivity of the questions. If so, the survey may be treated as defective.