How should you handle requests to speak with your current law firm when applying to new firms?

Answer: Good questions! First of all - and most important - never, never, never let someone speak to your current employer until:
 
  1. An offer of employment has been extended
  2. You have accepted that offer
  3. You have told the people at your firm that you have accepted an offer
  4. You have told the people at your firm that someone will be calling to check references.

There is absolutely no reason on earth that a potential employer should feel compelled to contact your firm prior to you accepting an offer of employment. You are absolutely correct in thinking that the partners at your firm would react negatively if they found out that you were interviewing.