Depends on what tax you're referring to. I'm assuming you're asking about income taxes, but there are other taxes an LLC may be required to pay. For example sales tax and state fees for having the privilege of doing business in that state. And there may be other taxes depending on your particular case.
In the case of sales tax and state fees you'll have to pay them whether your LLC is generating a profit or not. In the case of income tax, it's a bit more complicated, but generally you won't have to pay the state or US federal government if there is no profit.
An LLC can choose to be taxed as a partnership (pass-through taxation) or as a corporation:
Corporate Taxation: If your LLC is taxed as a corporation, then it is treated as a separate entity for tax purposes. That means that any profits generated by your LLC will stay in the LLC. The LLC will pay taxes on the profits, and you as the owner will only be taxed on the money you take out of the LLC. So, if your LLC didn't generate any profits, then it will not owe the IRS or the state any income tax because there was no income. But if you withdrew money from the LLC, you as an individual will owe the IRS and the state income tax on that amount whether or not your LLC had a profit.
Pass-through Taxation: If your LLC is taxed as a partnership, then its profits & losses are passed through to you as the owner. The LLC itself does not pay income taxes, you as the owner will pay any tax due as a result of profits generated by your LLC. This applies whether or not you withdraw money from the LLC. So, if the LLC generated no profits, then the $0 bottom line passes through to you as the owner. Since there is no income to report, you will not owe any income taxes. However, you may still owe both the federal and state governments money if you have income coming in from another source. For example if you have another job.
I hope this wasn't too confusing. The bottom line: An entity, this can be an individual or a business, generally will not owe the state or US federal government income tax if there is no income. Here are a couple of IRS resources that should help you:
I can tell you what stands for New York state.
In a LLC you file taxes together with your regular income taxes period.
If I recall correctly, my accountant told me that a LLC has to pay a minimum tax to the state regardless revenue. New York state used to charge an approximate $1000 per fiscal year. Now that amount is lowered considerably.
Your best bet is to ask your accountant what works in your state.