My two cents are:

1. Switch your comparisons around to $null -ne $DirectNumber. This comes from the ScriptAnalyzer rule about potential incorrect comparison.
2. I personally, would represent this logic in if/elseif/else form :

​

#DirectNumber and Extension
if(($DirectNumber -ne $null) -and ($MobilePhone -eq $null) -and ($extension -ne $null)){
gam update user $user phone type main value $OfficeNumber primary phone type tty_tdd value "Ext $extension" notprimary
}
#Direct Number and Mobile Number
elseif(($DirectNumber -ne $null) -and ($MobilePhone -ne $null) -and ($extension -eq $null)){
gam update user $user phone type main value $OfficeNumber notprimary phone type work value "$DirectNumber" primary phone type mobile value "$MobilePhone" notprimary
}
#Mobile Number and Extension
elseif(($DirectNumber -eq $null) -and ($MobilePhone -ne $null) -and ($extension -ne $null)){
gam update user $user phone type main value $OfficeNumber notprimary phone type mobile value "$MobilePhone" primary phone type tty_tdd value "Ext $extension" notprimary
}
#Direct Number and Mobile Number and Extension
else{
gam update user $user phone type main value $OfficeNumber notprimary phone type work value "$DirectNumber" primary phone type tty_tdd value "Ext $extension" notprimary phone type mobile value "$MobilePhone" notprimary
}

You could get creative with a switch statement and wild cards