He is looking forward to starting his job next week.
To there is a preposition that normally requires a noun as its complement. It can also take a gerund (verb+ing) because starting(gerund) has noun properties. Still, it's a verb and that's the reason why it can take "his job" as its own object.
He is looking forward to start his job next week.
This one is ungrammatical. Start is a regular verb and doesn't have noun properties like starting.
But start is a noun if preceded by a determiner like the and takes an of phrase complement as exemplified in Andrew Leach's example.
He's looking forward to the start of his job next week.