The death of Robin Williams got me curious about it, because as a person suffering from depression, hearing about his suicide made it very difficult for the next day to keep suicide off my mind. I was just curious if there has ever been a noted incident of the suicide rate correlating with celebrity death.
Definitely. This is a fairly well-known phenomenon, called 'suicide contagion' or the Werther effect, named as such because of the spike in jilted lovers' suicides following the publication of The Sorrows of Young Werther. If you have JSTOR access, here is a paper describing the effect.
In this specific case, there've been a few events, most notably the AMPAS's tweet saying, "Genie, you're free," that have triggered a real backlash from public health officials and people involved in treating mental illness, because it's precisely that sort of coverage that can contribute to a rise in suicides. In contrast, when Kurt Cobain passed away, medical professionals worked with the media to get across the message of how terrible the event was for his family and friends, as well as providing information about how to get help if one was suffering from depression, and the suicide rate in the region actually dropped.
With more awareness and responsibility among the media and others, it's possible that this effect can be successfully ameliorated or eliminated entirely, so hopefully in the future such tragedies will not have to be followed by others.
I can't speak of rates. However, the death of Rudolph Valentino comes to mind. He died of sudden medical issues, perhaps involving drug use, in 1926. There were widespread reports of mass hysteria and reports that many of his fans committed suicide following his death. Rudolph Valentino was sort of the Marilyn Monroe of his time, in terms of a sex symbol actor with a sudden death. I'm only on mobile, so my investigation is somewhat hampered in finding solid sources who evaluate the claim.
http://publicdomainreview.org/collections/rudolph-valentino-1923/