How can I install a 64-bit OS on a 2006 Mac Pro with 32-bit EFI?
There are instructions online that tell you how to replace the 32-bit EFI with a 64-bit version. Obviously this comes with certain risks and certain issues. Worth pointing out that I haven't tried this myself.

http://lowendmac.com/2014/modernizing-the-mac-pro-11-and-21/

If you want to install 64-bit versions of Linux there are a number of resources online that explain how to accomplish this. For example the following page has links to Linux install images that have been specifically modified to allow installation on 64-bit Macs with 32-bit EFI.

https://mattgadient.com/2016/07/11/linux-dvd-images-and-how-to-for-32-bit-efi-macs-late-2006-models/