What will be closed in Dresden/Erfurt on Corpus Cristi Day?

The only things that usually close on German holidays are the stores. These days are quite popular for families to go to museums, art galleries, restaurants, and other attractions, like pools, etc.

Train schedules will slow down, but depending on the popularity of the train, not that much. Subways that would normally run every 20 min. will cut back to once an hour, ones that run ever 5 min. will cut back to 10-15 min. Rural buses, may cut back considerably, especially those that are heavily used for kids going to school. Intercity trains may cut back too, so the best thing to do is check the train schedules for those dates.

Holiday schedule most often follow the Sunday schedule, so a holiday is not that big of a deal.

Nothing will be closed: Corpus Christi isn't a public holiday in neither Dresden nor Erfurt. In both states (Saxony and Thuringia) Corpus Christi is only a public holiday in municipalities with a roman-catholic population majority. Dresden and Erfurt are protestant.