When you breach a contract, you can get sued in local court, and if you don't show up to defend yourself, default judgment will be entered against you. Then the aggrieved party will have to collect, but the court in Washington (to invent a jurisdiction) can't enforce an order against a person in Norway (to invent another jurisdiction). So the aggrieved party would need to take enforcement of the judgment to the Norwegian courts. In the actual case of Norway, this is fairly simple, you just call an attorney in Norway to do the paperwork. It might be harder if the other jurisdiction is Belarus.
If you return to the US, even if there is a money judgment against you for the rent owed, you will not be arrested for that debt. Depending on the state (about half of the states), you might be arrested for failing to comply with a court order to pay the debt. The difference lies in refusing to comply with a court order, versus simply having a debt.
The State Department conveniently lists the reasons for denying a visa. Owing money or having an uncollected judgment against you is not one of the possible reasons, in fact even having been ordered by a court to pay, ignoring the order, and the court issuing an arrest warrant does not make you inadmissible.
