If only one of the four owners of a house lives there is he liable for the entire property tax?

In theory, all owners of the property are responsible for the property tax and it is up to them to agree how to divide the obligation among themselves. The tax authorities will post a tax lien on the property (regardless of who owns it) if the taxes are not paid, and attempt to notify the last known owner of record prior to taking away the property to cover unpaid taxes.

As between the owners, if one of them pays all the taxes (to keep the property out of "hock"), it becomes a "lien" against the ownership interest of the others, meaning they will have to pay their agreed share or otherwise fair share once the non-payers' property interests are transferred.