If we received a subpoena duces tecum for deposition - what would happen if we don't go?

If you don't show up you may be held in Contempt of Court.

A subpoena duces tecum pertains to the production of documents or things from a third party. It may be part of a subpoena to appear for a deposition, so the deponent would bring documents to the deposition. If you were only recently served, it may be possible to object to the production and/or the deposition. But the statement that you "already have a scheduled court date next month for the same thing" seems more consistent with a party-in-suit, rather than a third party. In any event, the fact that there is to be a court hearing in the same case, generally is not an excuse to ignore a subpoena. Failure to attend without following proper procedures could be grounds for seeking sanctions from the court. You should consult with a litigation attorney about the specific facts and procedures for your case.

If you believe that you have a valid legal reason for refusing to appear, or for refusing to produce certain documents which are requested "duces tecum" then you need to file an appropriate motion with the court and have it heard by the Court before that date to the extent it is possible. If you just simply refuse to appear, however, you may be held in contempt of Court. (Usually, when a witness fails to appear for deposition a civil matter pending in Florida state court, the party seeking the deposition first files a Motion for Order to Show Cause. However, a subpoena is a court order, even if it is issued by the lawyer, which must be complied with, so you should either appear or file an appropriate motion as to why you should not have to appear and have it heard before the date the appearance is required.) Finally, if it is just a scheduling issue, usually if you call the lawyer far enough in advance he or she will move the date for you as a courtesy. Make sure you have any agreement as to scheduling in writing.

If you fail to respond with the requested documents the party seeking the deposition can ask for you to be penalized, sanctioned or held in contempt. If you are unable to respond, then you should indicate the reason, in writing to the proponent of the deposition or subpoena. Ignoring it is unwise.