What are the pros and cons of dealing with a very large dataset in qualitative research?
There are no pros or cons just as there are no pros or cons to having $1,000,000 versus $1,000 to work with.  You are less constrained.  That is just a fact. The range of things you can do goes way up.  If you don't like things being larger, you can voluntarily choose for them to be smaller by sampling.  If you have a more specific question regarding methods, you should ask it on Cross Validated.