After spending time looking at group policy, as far as I could tell, there was nothing that was relevant. I then came across this post that suggest I try launching a command promt as an administrator and running

msiexec /a install.msi
This appeared to work, but ran very quickly - in fact it didn't. On a whim, I tried this inside the admin command prompt.

msiexec /i install.msi
which worked a treat.

I've run into this behavior, where MSI installation is prevented with "The system administrator has set policies to prevent this installation" before. I believe it is due to default Windows Software Restriction policy and I've seen it on both Windows Server 2008 R2 and Windows Server 2012.

There is probably a better GUI based way to alter the policy, but setting the following reg key as an admin on the machine does the trick:

[HKEY_LOCAL_MACHINE\SOFTWARE\Policies\Microsoft\Windows\Installer]
"DisableMSI"=dword:00000000

This is usually due to a Group Policy or a Local Policy disabling Windows Installer.

Look on your Domain Controller if you have a GPO disabling "Windows Installer": Computer Configuration > Administrative Templates > Windows Components > Windows Installer > "Turn off Windows Installer".

Verify that it is set to "Not configured" or "Disabled".

If if you don't find it in your GPO's, look on the computer's local policies using gpedit.msc.

Note: the fact that you are a domain admins doesn't matter. It all depends of the scope of the GPO.

right-click -> run as administrator

Neither the msiexec nor registry fixes worked for me (even after rebooting), but disabling User Account Control and rebooting did the trick.

You have the option to turn off UAC via registry by changing the DWORD EnableLUA from 1 to 0 in HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\policies\system. You will get a notification that a reboot is required. After the reboot, UAC is disabled.