Is there a way to find out what printers a user has mapped remotely?

Well, you can take a look at the Win32_Printer WMI class,

Get-WMIObject Win32_Printer -ComputerName $Comp

But, I think this will get you even better results:

New-PSSession $Comp | Enter-PSSession
Get-ChildItem Registry::\HKEY_Users\$UserSID\Printers\Connections
Exit

Without using PS Remoting, you could do this instead:

$Printers = [Microsoft.Win32.RegistryKey]::OpenRemoteBaseKey(Microsoft.Win32.RegistryHive]::Users, $ServerName)

You get the idea. Basically, you need to access that user's registry key with whatever method and enumerate whatever you find in Printers\Connections.

For anyone that's interested in what working code looks like to enumerate the accounts and get a listing of the installed printers, please see below:

Get-ChildItem Registry::\HKEY_Users | 
Where-Object { $_.PSChildName -NotMatch ".DEFAULT|S-1-5-18|S-1-5-19|S-1-5-20|_Classes" } | 
Select-Object -ExpandProperty PSChildName | 
ForEach-Object { Get-ChildItem Registry::\HKEY_Users\$_\Printers\Connections -Recurse | Select-Object -ExpandProperty Name }

This snippet first enumerates all subkeys under HKEY_Users, it then filters out the default/system account keys and the Classes keys for each user, finally it enumerates each remaining key's \Printers\Connections subkeys to output the printer information to the console.

Kudos to Ryan, so thought I would contribute via an answer.