I figured this out.
This works with my 1st gen Kindle Fire and my Mac running OS X.
First, notice that the Kindle Fire mounts as the volume "KINDLE".
Second, realize that there is a file on your OS X installation called /etc/private/fstab which controls filesystem mounting.
Third, use the command
sudo vifs in Terminal or otherwise edit the /etc/private/fstab file and insert the following lines:
# Kindle Fire LABEL=KINDLE none msdos rw,noauto
Fourth, try plugging in your Kindle Fire via USB. It should not auto-mount anymore.
Bonus: this works for any Android device that auto-mounts when you plug it in. My new HTC One (Verizon 32GB) auto-mounts HTC Sync Manager. It uses the label "CDROM". I added a similar line above, but substituting "CDROM" for "KINDLE" and now it stops prompting me to install HTC Sync Manager. In this case, I inserted these lines into /etc/private/fstab
# HTC One LABEL=CDROM none hfs rw,noauto