- Objecting to the Congressional Hearings on "The Radicalization of Muslims"
- Firing of Juan Williams from NPR for "offensive" comments towards Muslims.
- Support the "religious right" to build a Mosque near Ground-Zero of 9/11.
- Walk off of their own show (The View) in defense of Muslims 
- Call to not jump to conclusions on Fort Hood shootings 
|Liberal Democrats||Muslims and Sharia Law|
|Believe in separation of church and state||Believe religious law should BE state law (Sharia Law)|
|Believe that religion should not be forced on anyone||Becoming non-religious, is strictly forbidden and punishable by death|
|Believe in equal rights for women||The rights of women are arguably fewer. Polygamy is legal.|
|Believe in criminal rights and reform||Penalty for theft is cutting off one or both hands|
|Believe in sexual freedom||The punishment for adultery is stoning|
|Believe in equal rights for homosexuals||Homosexuality is illegal and punishable by death|
|Against death penalty||Death penalty for adultery, homosexuality, apostasy, etc.|
|Believe in legalization of controlled substances such as alcohol and marijuana||Liquor and drugs such as marijuana, cocaine, opium, etc are all unlawful|
In all fairness, these are extreme examples and the beliefs are not necessarily shared by all Muslims. Also, this list excludes any similarities(for example, Muslims allow for abortion up to 4 months of pregnancy). Still, the differences are significant.
The best explanation that I can come up with for the love affair is "The Enemy of my Enemy is my Friend" or in other words, liberal democrats think "Muslims are the enemy of Christians, who are typically conservative Republicans, therefore we like them". It's a little like teaming up with Stalin to beat the Nazis - after you get rid of the Nazis, how do you get rid of Stalin?
Update March 15, 2011 - Another possible explanation is that liberal democrats always look out for the underdog; even if they are ideological polar opposites.