Let's say I have a monthly mortgage payment of $1600 on a condo I own, and the condo is rented out for $1600. Then, I'm breaking even. Hence, I would not need to pay any tax on my rental income. Would that be correct?
Some of your mortgage payment is interest, and some is repayment of principal. When it comes to deducting your rental property's expenses against its income, you generally aren't permitted to deduct the repayment of loan principal.
So, no, $1600 rent and $1600 mortgage payment (ignoring all else) is not break-even — rather, you are profiting to the extent that your rental property's mortgage principal is being reduced. That profit is taxable income, even if your cash flow hasn't benefited. Otherwise, you could "break even" this way in perpetuity and at the end of the mortgage have a property paid for fully in tax-free dollars! Nope.
(I said "generally" at top because the possibility of an interest-only loan does make the entire payment interest, but I mention this merely for completeness and not because it is likely to apply.)