Welcome to the Wefugees community and thank you for contributing to our platform :-)
First of all: if your boyfriend applied for asylum before, Germany would unfortunately recognize the other EU-country (in that case, Italy) as the responsible country for his asylum claim and request a "transfer" back there. Have a look at this thread from our platform for more information on Dublin: Does Dublin III effect me if I seek asylum in Italy?
Getting married is a separate process (so even if he was rejected, marrying you gives him the right to a residence permit in Germany). Please note that having a passport is one of the core requirements in order to get married in Germany.
Here is an explanation and a list of what documents you need from our Wefugees Blog article "How to Marry": https://wefugees.wordpress.com/2018/04/20/heiraten-in-deutschland-how-to-marry/
You can also have a look at our platform for more information on the marriage process. To do so, simply use our search bar on the right side!
I hope this helps. Feel free to reach out to the community again if you have any more questions.
All the best and Good Luck,