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 a preposition can generally license a gerundgenerally license a gerund**(*** i.e. starting *) as opposed to other classes of verbs.( A gerund is close to a noun). Still, it'sstarting is a verb and that's the reason why it can take "his job"his job as its own object.
He is looking forward to start his job next week.
This one is ungrammatical. Preposition to cannot license this form of verb.
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.