The poor people in many cases aren't paying the rents directly. The local governments are renting places at or above market rates to house poor people.
But on the whole the rising rents in Berlin is more due to the influx of European tech workers and the like, rather than refugees.
(Subsidized housing as in: The government subsidizes your housing project if you agree to fixed rents for a certain period of time, afaik 20 years or so.)
- highly skilled workers (IT, Finance, Sales, Marketing) that earn much more than average german folk, and compete for housing with upper class berliners
- unskilled workers (constructions) that live usually in temp housing and don't attempt to rent a flat for themselves.
As for the refugees, most of them still live in camps distributed across Germany and if they get a flat subsidised by the state it will be in some low demand neighbourhood.
Seriously though in the UK we have had immigration over the last 20 years and that must have led to higher demand and prices. Best estimates are about 20% in 20 years in England.
https://fullfact.org/immigration/have-house-prices-risen-bec...
BUT
English house prices have risen by 320% on average over this period.
Other factors are - getting married later, more single people, more divorced couples, and financialisation of housing.
Probably a similar story in Berlin.
However, since the cheap money from the banks, more people are orienting to buy than rent, which is quite different from the past.