As far as I know, no jurisdiction in the US relieves a person of their contractual lease obligations when they are required by law to leave the country. However, many (perhaps most) states require the landlord to make an effort to re-rent an abandoned unit, which reduces the size of the tenant's liability. Unless the landlord just gives up on the claim for $8K, you would have that liability, which could be collected through legal process. The landlord might do this in her jurisdiction, which would no doubt result in a default judgment (because you won't show up). Theoretically, she could also pursue you in your home country, but that is much more complicated, expensive, and subject to whatever discretionary dim view the local courts have of US immigration law, or other tenant-related laws (the courts would have to conclude that there was a wrong by the standards of that country). A variant of suing you in your home country is suing you in the US and having the judgment enforced in the foreign country, but that against depends on the legal system of that country and treaties between the country and the US.
Since visitor visas are a highly discretionary legal entitlement, it's impossible to say "won't" w.r.t. the question of whether a visa application will be denied because of an unpaid monetary judgment. It's also impossible to say "won't" because the law can change at any time. With those caveats, there does not appear to be any concrete reason to believe that a visitor visa will be denied because you skipped out on a lease. A visa can be denied and one can be perma-banned for lying on interview questions, so the hope is that there isn't a question "do you have any money judgments against you in the US".