USPS actually writes checks to pay taxing authorities. That way you don't run into the problem of a property owner neglecting to pay those taxes and the USPS finding out later that the sheriff wants to seize the building for a tax sale, and all that sort of stuff.
The USPS definitely would not have its operations hindered by a property owner neglecting to pay his property taxes. First, the seizure and sale would not affect the lease under the law of any state; in my home state of Louisiana, the result would be something called a "real subrogation," which means that the lease continues as is, just with a new entity collecting the rent checks.