The case against doing so:

Mayo Clinic psychiatrist J. Michael Bostwick recently addressed this subject (pdf):

Just because the general public wants to believe the tautology that heinous crimes must be the province of the mentally ill (because no one in his right mind would perpetrate such acts) does not make it so.

In a nationwide Swedish study of 13 years of violent crimes such as homicide, aggravated assault, and robbery, individuals discharged from psychiatric hospitals with severe psychotic or affective diagnoses did have 3.8 times the odds of committing such crimes than did their none mentally ill countrymen. However, their number relative to the general populace was so low that only 1 in 20 violent crimes could be attributed to them. These findings are consistent with earlier American studies, which estimated a 2- to 4-fold increase in the risk of violence by individuals with schizophrenia but only a 3% to 5% population-attributable risk.

Calling the epidemiology of mass murder “counterintuitive,” Friedman and Michels write that “we must explain an epidemiologic fact that the public likely finds counterintuitive in the wake of a mass killing: Although mass murderers probably have more psychopathology than other killers, the mentally ill as a group pose little risk of violence.” Moreover, Appelbaum warns that increased violence may not actually be a result of the mental illness itself but of comorbid substance abuse and sociopathic personality traits. Given these statistics, the American Psychiatric Association has questioned both the “fundamental fairness” of re-stricting firearm access for the mentally ill and the possibility that such restrictions could further stigmatize an already marginalized group.