From https://metaculus.com/questions/17100/us-api-restrictions-on-ai-before-2026/

As of May 2023, US law does not restrict API or browser access to any Large Language Models (LLMs), such as GPT-4 (ChatGPT) or Google Bard. As the capabilities of such systems grow, and users learn more how to extract certain types of knowledge (e.g. how to enrich uranium or create Deep Fakes), one way US policymakers might address risks would be to restrict usage to those that are verified to be in the US.

A similar regime exists for Digital Rights Management for music & movies - access to music on Spotify or shows on Netflix or Prime Video may depend on your country of origin. (And people regularly use VPNs to pretend to be from some other place to access this content.) Copyrights and trade agreements more generally often have provisions for various countries, and some countries like China ban access to entire systems via firewalls.

This question asks about APIs, not browser or app usage, because APIs are the most flexible way to use large AI systems. (In April 2023, Italy banned and then unbanned ChatGPT's interface but not the API, which was decried as ineffectual.)