To demonstrate it to visitors, you need it to be free initially.
Unlike throwing up a normal Web site, which can be hosted almost for free, the GPT-4 part with lots of tokens could make free users expensive (even with only legit users just trying it out; nevermind broken bots or malicious attackers).
If it gets traction, you'd like to either charge subscriptions for the service as a small business, or get acquired, or get funding to go bigger.
From your kitchen table, using only handful of coins you hid in a jar... how do you bootstrap this side-project startup that depends on expensive LLM?
Self-host your own LLM (use one of the freely available ones, possibly including the leaked Facebook Llama stuff) and allow unlimited free experiments / tests / whatever with that. Offer a (very?) small number of free experiments / tests / whatever against the live OpenAI API. Support paid subscriptions from day 0 for people who really want to jump in and do a lot of heavy stuff with GPT-4.
Another thought is to record a screencast of yourself running some demos against the live OpenAI API and put that video up on the page instead of, or in addition to, assuming that the site itself has to be free initially.