I too keep my Calibre library in Dropbox, and I read epubs on my IPad Mini 2. You need an IOS ereader app that can download your ebooks from Dropbox. I have experience of two such apps:
Marvin. This has a nice, user friendly Dropbox interface. Unfortunately Marvin is no longer supported by the developer.
Mapleread. Also interfaces to Dropbox, but this interface is not quite as user friendly as Marvin. The rest of MapleRead is good, and it has good developer support. This is my current e-reader app.
Once I add a book to Calibre, it automatically syncs to Dropbox in the cloud - I assume you will be familiar with that. I then close Calibre before going into MapleRead on my IPad, and wirelessly downloading the new book, ready for reading.
Apparently sharing your Calibre library via a cloud service like Dropbox can cause problems, but I have never had an issue with it. I make sure to always close Calibre first, and wait for Dropbox to finish any syncing, before attempting to download into MapleRead on my IPAd.
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