In a rented house who’s responsible to put up mezuzot? If it is the Tenant then what is the halacha of removing them when the tenant leaves? All outside of Israel.
The tenant is responsible for the mezuzos, because the obligation of mezuzah applies to the person living in the house.
After the tenant moves out, if the next tenant is Jewish, there is a problem of taking down the mezuzos while the house remains empty, and it is therefore best to come to an arrangement with the landlord of with the next tenants that they should but the mezuzos from you after you leave.
If this doesn’t work out, it is permitted to switch the mezuzos for cheap (kosher) mezuzos, and take the dearer, more mehudar mezuzos with you.
See Shulchan Aruch, Yoreh De’ah 291:2.