Finally a question I can answer!
No you do not have to remove your tanks. Turn them off as normal and think of it this way, your gas cannot get out of the tanks and the fumigant cannot get in. You should be provided with "special" bags to place all ingestables and medicines in. The fumigation company is most likely going to use a fumigant called Vikane it will not harm your torch but leave it in the off position with the hoses connected.
The natural gas will be turned off to the building because the gas can react with heat (from pilot lights) causing a reaction.
Vikane does not leave a residue and is colorless, odorless and DEADLY to people, plants and animals. Chloropicrin (tear gas) is added to the property as a warning agent.
If you have any questions feel free to PM me, while the laws in Florida may vary from California I will try to help.