Some homeless people avoid shelters even if there are space because they are frequently giant nests of disease and violence. Besides catching some rather unpleasant diseases and pests, your odds of getting stabbed increase dramatically.
So you'd have to pay for hygiene, health, and security expenses well above and beyond the norm. And that's just if you want to pursue the "containment" side of housing the chronically homeless.
If you want to pursue the "rehabilitation" side it gets even crazier. A large portion of the population suffer from severe mental health problems that prevent them from any meaningful integration into mainstream society, and mental health care ain't cheap.
That said I do believe it should be done, but it isn't "oh yeah just levy a tiny tax" territory.