The law requires the tenant to insure himself against the risks for which he is responsible as a tenant, if he rents an unfurnished property as his principal residence.
For other types of leases (furnished, commercial lease...), the lease generally requires the tenant to take out insurance covering his risks.
For property in co-ownership, each co-owner is obliged to insure against the risks of civil liability for which he/she is responsible in his/her capacity as owner-occupier or non-occupier.
For other property, the owner is not obliged to take out insurance to cover damage to the property. However, in view of the low cost of such insurance, it seems reasonable to take it out.
The landlord 's insurance is generally called "PNO" (Propriétaire Non Occupant) insurance. It may be useful to take out a guarantee that fully covers the property during periods when it is not occupied by a tenant.