I got a notice from IRS notifying me of penalty for late filing. Business income was zero and is closed. Am I personally liable?

If the notice was to a business entity, then you probably do not have personal liability. However, if the business was operated as a sole proprietorship, you probably do have personal liability. Contact a competent CPA to discuss this.

Most penalties for late filing are based on the tax due. You don't say what kind of tax return this was or what the penalty was based upon. A business can have employees and still have no profit but still be required to file 941 and pay over payroll taxes withheld. If the business filed 941 late and paid the payroll taxes late, there could very well be penalties and if you are the owner or responsible person you could be found 100% liable. You don't say what the form of the business was (Corporation, LLC, Partnership, Sole Proprietorship) or what your relationship was to the business. You should schedule an appointment with a tax attorney to answer these question (after you provide all the information needed.

You can not have a penalty for late filing with the I.R.S. unless there is a balance owed for past due taxes. You indicate that the business income was zero, but the fact that a late filing penalty was assessed suggests that there was income reported to the I.R.S. I say this because the late filing penalty is actually a percentage of the tax owed, and you can't have tax owed without income. You should order the wage and income transcripts from the I.R.S. for the business, or otherwise try to determine if any income was reported to the I.R.S. for the business.