Dear A Rahman,
Welcome to the Wefugees Community and thank you for reaching out to us.
The platform and our experts are based in Germany. Therefore, we are not always able to provide profound information or to share experience regarding regulations and procedures in other countries.
Nevertheless, a very similar question was asked here before and you may want to follow this link for more details. According to this website, recognized refugees as well as asylum-seekers are in general allowed to marry in Greece. The most problematic part, however, is to be able to provide all the required documents. Asylum seekers need to consider, that getting in touch with the country of origin's embassy might jeopardize the asylum claim. You mentioned that your boyfriend's asylum application got rejected. In case he appealed this decision, the same might be valid for him.
Apart from the technical and procedural questions, you are only 16 years old and (as far as I could find out) according to UK-law not allowed to get married (at least without the permission of your parents or a court). From my very personal point of view, I hope that you can find another way to support your partner in Greece, as you are still very young and a marriage is quite a big and drastic step.
In any case, please don't hesitate to get back to us with any further questions.
All the best,