1) It will have an enormous cost. TfL in London collects £5bn/yr in fare revenue. While fares are lower in Berlin and it's a less busy system, it will still probably cost €500mn/yr for Berlin alone.
2) Most people driving are not doing it because they cannot afford public transit fares. They are either wanting a more 'luxurious' environment, can't get where they are going easily on public transit, etc. I'm not convinced it would actually induce much demand on the system from car drivers.
3) It may induce a load of demand from non-car drivers, who can now do a load more trips, longer commutes, etc.
4) Most public transit systems are at peak capacity in rush hour. In London if you switched everyone who was driving to public transit you'd need to double the capacity to do so. Considering crossrail is being built for ~£20bn to increase capacity 10%, you'd probably be needing to spend hundreds of billions of euro on capacity upgrades.
Much better to actually tax the diesel cars properly for the externality cost rather than do some other policy which doesn't directly address the issue.