I have set lavatory sinks in a few old dressers as antique period looking vanities, some are easier to modify than others. What you are thinking of can probably be done and could make a very nice furniture style chamber. How to go about it will depend on the way the inside is constructed. Pictures would help.
You don't want to cut so much that it isn't structurally sound. What you can safely cut will depend on the way it was put together.