It isn't that symbolic, actually, but a set of meanings barely beneath the surface. This is Anthony talking to Cleopatra. These are famous lovers. He is Rome to her Egypt, a land related to and defined by its rivers. Most simply he is saying, I don't care—let huge amounts of time pass, and he won't do anything, because he doesn't care. He just wants to hold Cleopatra. But we also talk about passion and lovers being hot, so he's saying things are hot—and perhaps that his Roman firmness is going to "melt" into her (they will have sex, and his erection will melt).
It's not symbolic however, Anotny used this phrase in front of Cleopatra to make her happy. Cleopatra was not happy with the arrival of the messanger from Rome so, Antory says these lines which means that he no longer worries about Rome and let is drown in the river. This shows his negligence from Antony's part as a General, and being one of the triumvirs this doesn't befits his character. He was so busy in love making and in pleasure with Cleopatra that he no longer cares for anything.
This novel is based up the lovers Antony and Cleopatra. Antony one of the triumvirs starts living with Cleopatra neglecting his responsibilities as a general. When the messenger came to inform him about the War he replied to the messenger 'let Rome in Teiber melt' which shows that he doesn't care whatever is happening in the Rome and he wants to live pleasurable life with Cleopatra. In a sense, this phrase has a symbolic meaning. It shows Antony's negligence as a general and Antory by saying this shows his love for Cleopatra.....