Germany has always had very low home ownership rates. The influx of migrants has put some pressure on rents (esp on the low end of the spectrum) but hasn't caused the surge in prices for houses or luxury flats. That was mostly driven by a booming job market, higher wages and low interest rates.
Berlin was always very cheap compared to other German cities. What we've seen now is more a normalisation than a bubble. But it certainly took some pressure of other markets that already were at very high prices.
Luckily, zoning laws in Germany allow new supply and there's a lot of that coming in soon. Some economists expect prices to fall by up to 20-30% in some areas over the next years as supply comes in and interest rates go up (likely starting 2019). That would probably be a healthy correction, easing the tensions that have developed.
An exception is high rents for flatshares. Not all flats are suitable for that so that an influx of young people (e.g. students) can increase prices for flatshares much more than that of other flats.
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.
This seems to be a possible hypothesis. Could you validate it?