Yes, you can but only with Intel Xeon E53xx, X53xx or L53xx series processors that have a LGA 771 socket.
Although one word of warning, the supported OS upgrade life time of the Mac Pro 1,1 and 2,1 looks like its coming to an end. Apple as we have seen in the news is prepping for Mountain Lion, and dropping support for all macs that lack a 64bit EFI. So it appears highly likely that Mac OS X 10.7 is the last OS you will be able to run on these Mac Pros, unfortunately. Be aware of that when prepping for your Mac Pro 1,1 and Mac Pro 2,1 upgrade plans, especially if keeping current is a priority of yours.
That said, now on to the upgrades:
The Mac Pro original Upgrade process
Here are processor upgrades that people have completed and confirmed in Geekbench as working, see their results below.
MacPros using L53xx Xeon processors.
MacPros using E53xx Xeon processors.
MacPros using X53xx Xeon processors.
Edit
As pointed out by Kent, an iMac with a 2.16 Ghz Intel 2 Duo seems to be like an iMac (Late 2006) instead of an iMac (Mid 2007).
Technical Specs for iMac (Late 2006)
Technical Specs for iMac (Late 2007)
Therefore, you can only run Lion. Unfortunately, finding a legitimate copy of Lion could be a little tricky if you haven’t already purchased it. If you search for OS X Lion in the Mac App Store you’ll only find Mountain Lion, and Apple no longer sells an OS X Lion USB Thumb Drive (part number MD256Z/A) in its online store.
One of your options is to buy from Amazon or another online retailer.
Or, get Snow Leopard.
Original Post
You have to install Snow Leopard first and then Mountain Lion.
OS X Mountain Lion system requirements
To install Mountain Lion, you need one of these Macs:
**iMac (Mid 2007 or newer)** (Yours)
MacBook (Late 2008 Aluminum, or Early 2009 or newer)
MacBook Pro (Mid/Late 2007 or newer)
MacBook Air (Late 2008 or newer)
Mac mini (Early 2009 or newer)
Mac Pro (Early 2008 or newer)
Xserve (Early 2009)
Your Mac needs:
OS X v10.6.8 or OS X Lion already installed
2 GB or more of memory
8 GB or more of available space
Your model must have a 64-bit EFI boot ROM.
An easy way to tell if you are running a K64 kernel is to use the uname command-line program. The "x86_64" in the excerpt below means that we are running a 64-bit kernel. If the output showed "i386" instead, that would mean a 32-bit kernel.
uname -a Darwin... root:xnu-1456.1.25~1/RELEASE_X86_64 x86_64
Therefore, you need to upgrade to Snow Leopard first.
Other Things
AirDrop and AirPlay will not work on your computer.
Sources
http://support.apple.com/kb/HT5444
Alternate Way
It is possible, but not necessarily legal, to install Mountain Lion over Leopard.
You can look at this.
Best Answer
According to the tech specs, you should be able to use OS X Lion (10.7) with the following specs:
That would suggest to me that you would need to get onto 10.6 (Snow Leopard) + all updates and then could get to OSX Lion (10.7).
Mountain Lion+ Specifies an "early 2008 Mac Pro or newer", yours appears to be the iteration just before that.