Solution 1:

This Microsoft article discusses this same issue:
Information about using 2007 Office suites and programs on a computer that is running another version of Office.

In general, Office 2007 can coexist with earlier versions, but Oulook apparently cannot :

Multiple versions of Outlook

Microsoft Office Outlook 2007 cannot coexist with any earlier version of Microsoft Outlook. When you install Outlook 2007, the Setup program removes Microsoft Office Outlook 2003, Microsoft Outlook 2002, and Microsoft Outlook 2000. The Setup program removes these versions of Outlook even if you click to select the Keep these programs check box in the Removing Previous Versions dialog box.

Solution 2:

To add the same quote for Office 2010 as harrymc added for 2007:

Multiple versions of Outlook Microsoft Office Outlook 2010 cannot coexist with any earlier version of Microsoft Outlook. When you install Outlook 2010, the Setup program removes Microsoft Office Outlook 2007, Microsoft Office Outlook 2003, and Microsoft Outlook 2002. The Setup program removes these versions of Outlook even if you click to select the Keep these programs check box in the Removing Previous Versions dialog box.

Solution 3:

You would need to use a virtual machine. Something like Virtualbox does the trick. Install the OS inside Virtualbox then clone it for each Outlook version and install your Outlook versions on each VM. I use this trick for html email testing

Solution 4:

Here is the in-depth reason why you cannot run any Outlook versions side-by-side.

Essentially, it's a "last one wins model" where the last COM component registered will be activated.