Drop dime means to rat someone out, or reveal information, in any situation. It derived from late 1960's and 70's South, where there were 33 fires a day at one point, and crime was the highest it ever was. Thus, many people would rat to save thier own ass, and they'd use a dime because that's what it cost.
Yo man, somebody dropped dime on me. My probation officer know about my bid'ness.