What is the typical consequences if a foreign person stop paying landlord rent for housing and leave USA?
This question is for a friend.
The law only addresses possible legal consequences, from which you could surmise what actually happens. If a person stops paying rent, the landlord institutes eviction proceedings, and may sue for the remainder of the rent for the term of the lease: they have an obligation to try to find a replacement tenant, to mitigate the damage. Some relief is available to the landlord in the deposit, from which the unpaid rent might be deducted, but let's assume that after that, the actual damage suffered is $5,000.
When you sue a person in your home state, the court serves official notice that the defendant must appear to answer the charges, and the state has jurisdiction over the respondent because they are in that state. Giving notice and collecting on a judgment across jurisdictions can get complicated and expensive, so it would depend on what kind of treaties exist between the countries. There is the Hague Service Convention which may simplify the business of giving notice to parties, which works if the respondent fled to Norway or India but not if they fled to Somalia or Turkmenistan. Since it turns out the US is not party to any multinational treaty recognizing foreign judgments, that country is not obligated to care about a US judgment, so unless the person left assets in the US, you may not be able to collect.
You could sue in the person's country (hire a local attorney to pursue the matter). The main legal question would be whether there is any conflict between the lease and local law: while the basic idea of a lease is universally recognized, there may be peculiar conditions or procedures applicable in that country. (Norway has some laws pertaining to "shared utilities" which don't exist in the US, which might put a US lease at odds with Norwegian law, and there are rent-raising rules that don't exist here). It might be necessary that you appear in court in that country to swear under oath that the lease was agreed to voluntarily, or the court may require a special form of 'notarization' not available in the US. None of that renders international litigation impossible, though a favorable judgment might be unlikely in some courts. The cost of litigation might be much higher than what would be awarded by a court.