OP did not write that rents would go up until l00% of people's income. IMO what he meant was that certain flats that used to be priced at the level that everyone could afford them, would go up in price when everyone's income would increase because of basic income: and they would keep going up until they reached the same equilibrium that they used to be in beforehand. Of course this would not just apply to rents but to all scarce commodities.

I think OP has a valid point; similar things happened after the influx of high income tech workers in SF, or for instance in some European countries after governments gave tax advantages to house owners.