Ultimately, the landlord and/or his Property Owners Liability insurer is responsible. If the tenant has Comprehensive motor insurance he can opt (at his discretion not the landlords) to have his car repaired under his comprehensive insurance, the tenants insurer will then reclaim their costs from the landlord or his insurer and the tenant would claim thier excess back from the landlord as well.
There is no legal requirement for the landlord to have Property Owners Liability insurance so I would always advise checking that the landlord has insurance or is in a position to pay before running up a huge bill particularly if there is no Comp cover in place!!
And with the shoe on the other foot - a landlord is well advised to ensure that he has Property Owners Liability on a policy where his business is accurately described as a Property Owner - otherwise he could find himself uninsured with a large claim on his hands particularly is someone is injured as a result o his negligence.