What I don't understand, though, is why Voldemort wouldn't say anything. Breaking Snape's ties to Lily specifically and the Light in general was obviously to his benefit.

I doubt Voldemort would have known who Snape was infatuated with specifically. Voldemort isn't the kind of fellow you swap feelings or share tales of old loves with. Snape is one of the best occlumens alive, it seems unlikely to accidentally give up his most secret memory. Doubly so if it's a mudblood. Triply so if it's a member of the Order of the Phoenix he's in love with.

Voldemort probably knew that Snape pined after someone (in a rather creepy and pathetic manor), but not specifically who he pined after. Well, not until he was on his way to kill her anyhow, and he hasn't had much opportunity to use it as leverage since then.