While I believe the Obama as a Muslim lie is part and parcel of the birther movement (the birthers nearly all also think Obama is a Muslim, or some variations of the smear is "Obama is a Kenyan Born Muslim" instead of just Kenyan born) some people treat them as different things. One common myth around the internets and even here is that Hillary Clinton started the rumor that President Obama is a Muslim. This follows as Trump and fellow Republicans in the media parrot this idea.
Thank God (or fate, whatever) for fact checkers. Politifact not only took on the assertion that Hillary and/or her campaign started the birther myth (rated: FALSE) but they have also now looked at the alleged connection of Hillary to "Obama is a Muslim" garbage:
Danielle Allen is director of the Edmond J. Safra Center for Ethics at Harvard University. Allen supported Obama, and in 2008, she tracked the origins of this claim to a perennial candidate and litigant in Illinois named Andy Martin in 2004. Allen said bloggers took some ideas from Martin and embellished them over the years. In October 2007, Politico reported that versions of the slur were circulating in South Carolina, apparently with zero connection to the Clinton campaign...
We could find no evidence of a deeper connection to Clinton or her campaign.
There you go! Hillary is not the reason for "Obama is a Muslim." That rumor had been going on as far back as 2004 near the time of Obama's DNC speech. This lie that she did need not be repeated on anywhere, especially a Democratic blog.