I lived legally in the US for 6 years on a student visa. I left 2 years ago, and due to family problems, I left without closing my company and without paying income tax. I want to come back and spend a week in New York. Would there be any problem entering, like immigration stopping me at the airport and sending me back?
Nothing is stopping you from paying your income tax.
You don't need to be in the US to pay income tax. Everything about it can be done online, or in the worst case by mail. So if you are concerned about this, getting your tax situation sorted out should be the first thing you do. Submit a late tax form and pay the bill. If it's a more complicated case then you may want to get a US-based lawyer involved. This can also be done without entering the US.
Not "closing" your company may also cause similar problems. The IRS may expect to see tax filings for the company for every year it is considered operating. You should sort this out while you are sorting out your personal tax.
There is one more problem you may encounter. In the general run of things people in the US on a student visa are not allowed to run companies. They can start and set up companies, but as soon as they actually do any work for the company (even unpaid work) they are violating the terms of their student visa. This is certainly true if they are on the usual F visa. This means you may have violated the terms of your visa during your last visit. There are some exceptions, and some ways round this restriction, and hopefully your case is one of the exceptions, but it makes sense to point this out in case you weren't aware of it. If you violated the terms of your student visa then you absolutely can be denied another visa, or turned back at the airport.
Sorting out your unfinished business from your last visit should be your first priority.