"continui" is "congiuntivo" and this sentence can be translate as "“let your brotherly love continue". The conjunctive can also express an order, an exhortation or a hope: Che nessuno si muova! = everybody stay still! Che qualcuno ci aiuti! = somebody help us! So "continuare" in italian also means to last or to keep doing something.

Thanks milena, which tense is used here?

"Continui" means continue; something be constant. In this case, continuare is inflected for lui form because of what has before (il vostro amore fraterno).