If you drop attributes entirely, all levels for that respondent are set to 0, thus the range of the attribute is 0 and it receives 0 importance. This would skew the average importance of an entire sample.
If you drop levels for respondents, you as the analyst have to tell HB what to do with the missing data. Choosing the option to set it to an arbitrary, large negative value would drastically inflate the range of that attribute, inflating its importance. Choosing the option to treat it as less desireable would make those levels have a lower utility than those that were kept, which is a good guess at their utility, but we don't quite know if it's the right range. Choosing the missing at random option, you aren't quite sure what you're going to get.
In general, we aren't huge fans of importance scores because they are directly affected by the range of levels you choose. If your current market has a very narrow price band, but your conjoint tests a wider band, you come up with X% importance of price, but that might or might not be how importance price is in real market decisions.
When attributes are dropped you could probably get around the skewing by grouping similar respondents and calculating group-level importances.