You need to have SOME kind of income to contribute to a Roth. I don't know the exact rules for what income qualifies, though. I'm pretty sure dividends and capital gains don't count. I don't know about rental income, though.
However, you only need to have as much income as you contribute--so, if you max out a Roth, you only need to have $5500 of income. So it's not like you need a ton of qualifying income. I'm just not sure about income that doesn't come from a W-2.