How much influence, if any, should religious organizations have in politics?

As private citizens they could say whatever they want, as organizations they should have 0% say on anything. I fully believe that government and any religion should be 100% separate.

The Constitution is clear on this matter, I think a more interesting question would be is should the government stay out of religious affairs?

Say? None. Input, just as much as anyone else. Prayer, I leave that to the individual and fully believe the Government has NO RIGHT banning or enforcing it anywhere. We are a nation that is based on a belief that there is a higher power. The modern Christian (zealot) believes it is his style of Christianity. Yet, if you study what the founding fathers debated and believed, the entire Bill of Rights and Constitution was prayed over, followed God's Laws, and is directly attributed to God.

When in the Course of human events it becomes necessary for one people to dissolve the political bands which have connected them with another and to assume among the powers of the earth, the separate and equal station to which the Laws of Nature and of Nature's God entitle them, a decent respect to the opinions of mankind requires that they should declare the causes which impel them to the separation.

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.

Among our founding fathers were enough differing ideas on religion that they, IMHO, boiled it down to a common core of belief. It is CLEARLY stated in the Declaration of Independance. It has both Christian and Deist wording. In fact, I am positive you could put on the glasses of any religion that adheres to "the golden rule" and see yourself in those opening words.

Sharia Law is not God's Law, it is Muhammed's Law, a prophet (as I understand it). Therefore it is a rule of man rather than a rule of law, and as such, does not belong here.

The whole idea of separation of church and state was to NEVER allow the church to rule the people, not even in the name of a King or body politic. It was NEVER intended to strip the individual's belief and practice while in office. What a gaggle we have let happen.

The First Amendment of the Constitution states that the Government shall make no law respecting the establishment of religion, i.e. no official religion of the government. It also states that there should be no prohibition on the exercise of religion. That is ALL that is said on "Separation of Church and State". That being said, organized religion, as a single entity, does not have an official say in the government. However, a person's expression of their personal thoughts and feelings cannot be suppressed when they are influenced by one's religious beliefs. That too, would be a violation of the First Amendment in regard to freedom of speech. Basically, individuals talking with government officials and expressing their view is okay, as is a representative of a group talking with their government representatives. The constituency of a government representative should be able to express to them their thoughts and views.

1 If they are a religious organization, then they should "keep the main thing the main thing".
2 That should be the priority but they still have a right and duty as citizens to hold those in elected office accountable to fulfill their duty. Whenever they start getting too far into rule #2, then they need to refer back to rule #1....

An opinion inspired by one's religious foundations, whatever that may be, is one thing; theocracy however is a whole different matter.

None at all to be consistent with the separation of church and state doctrine.

Absolutely zero. If they want to donate money to political causes, that is their right. If they want to protest political causes, that is their right. But they have no say in politics more than the vote that they cast for the individual they chose to represent them.

Using Catholics as an example... we've got the Pope. He is not an American citizen. He doesn't get a say in how we do things. Now as a political leader, it would benefit me to be on good terms with the Pope in order to also be on good terms with the vast amount of Catholics in our country. But that is the full extent of it. I am not obligated to do what the Pope demands any more than I am obligated to do what lobbyists demand. I might piss some people off, but it'd be my job to run the country, not theirs.

Next to none.

It depends on what type of involvement your are referring to. If its just voicing their opinion, donations or lobbying then they have just as much right as corporations or unions to have a say.