Certainly the rent in "poor neighborhoods" will go up, yes. Not sure the rents in my area will go up, because contrary to popular slashdot belief, almost nobody actually makes minimum wage.
If living at a poor neighborhood costs more, then people will start thinking about moving to better places that are still cheap. And that goes on and on and on. Most systems interact and housing prices are not exempt from that kind of price increase propagation.
In fact, most housing bubbles are created when credit lines are offered for low-income housing, just like the US Housing Bubble was created by subprime lending. People start thinking "If it costs X to live AT THAT UGLY PLACE, then my house is worth X + Y" and then the price goes up everywhere.