This phrase often implies the speaker thinks they're better than the person they're ignoring. Even though I tried to be friendly, the new manager didn't give me the time of day.
Pro Tip 2/3
Expressing Disdain
Use it to convey someone's disregard or lack of interest in another person. He's so arrogant that he doesn't give anyone the time of day unless they're famous.
Pro Tip 3/3
Indicating Importance
It implies the ignored person is deemed unworthy of attention or time by the ignorer. Since she won the award, she won't give her old friends the time of day.