It would violate the constitution. The federal government can't tax assets, but it can tax transactions (egs. earned income, sales, inheritance, dividends). Not paying rent is not a transaction, so the proposed tax would be an ad valorum tax on assets, which is constitutionally reserved for the states.

Not buying health insurance is not a transaction, either, but SCOTUS ruled it was constitutional to tax that.

That was a bit of trickery on the part of Zero. Congress has the power to levy taxes. They specifically can't levy a property tax and the states specifically can, whereas the states specifically can't levy a poll tax. So IIUC, in justifying this travesty of justice now called Obamacare, they levied a poll tax on everyone but waived it for those who had qualifying health insurance or who qualified for Medicaid. That way it's not a tax based on the value of property nor a specific transaction, it's a poll tax. But they did have to admit that it was a tax.

1HF