A self-refuting argument is one which must be false if it is true.
The excellent blog Tough Questions Answered has a list of self-refuting arguments: Are You Refuting Yourself?
“No English sentence is longer than three words.”
“There are no truths.”
“I do not exist.” The problem here is that a person must exist to make the statement that they do not exist.
“Anyone who is biased should not be trusted.” Isn’t the person who is making this statement biased himself?
“Only science gives us true knowledge.” How do you know that statement is true? It isn’t a statement of science.
“All truth is relative.” Is that truth relative?
“There are no absolutes.” Is that statement absolutely true?
“It’s true for you, but not for me.” Is that statement only true for you or is it true for everyone?
“You should be skeptical of everything.” Should we be skeptical of that statement?
“You can’t know anything about God.” Do you know that God is unknowable?
“You ought not judge.” Isn’t that a judgment?
“You should be tolerant.” Aren’t you being intolerant of me?
Some of these are commonly used by atheists/agnostics/skeptics (e.g. “Only science gives us true knowledge”) but I see them more commonly coming out of postmodernism. Of course, one can have a postmodern atheist, and that person can pick and choose which self-refuting arguments to use.
This doesn’t seem to be a logical mistake that Christian apologists make all that often.
Grace and Peace
GeoChristian 
In my experience Christian apologists don’t make self refuting statements but they do try to win points by using the strict rules of logic against anyone who makes even a casual statement.
A couple of examples from your list
You should be sceptical of everything.
Is this using ‘sceptical’ in its everyday meaning ‘given to suspending judgement, inclined to question the truth of facts and inferences’?
If so, the answer is ‘yes, we should be sceptical of that statement’.
In strict logic, sceptics were adherents of Pyrrhonism and denied the possibility of knowledge. If this is how you are using the word then the statement does become self refuting.
You should be tolerant. This isn’t self refuting as such but it is a favourite of the fundamentalists. Again it depends on how you understand ‘tolerant’. The fundamentalist Christian argument goes something like this.
Our viewpoint is that there is only one true viewpoint and the is one we hold. If you claim to be tolerant you are saying that you accept the right of others to hold any position. Thus you define our position that there is only one true position as intolerant. In fact you are being intolerant because you do not accept all viewpoints. You are excluding ours.
This a deliberate narrowing of ‘tolerate’. I do tolerate the fundamentalist viewpoint although I disagree with it, but the fundamentalist view is that anyone who is not for them is against them and by disagreeing I am not tolerating their position.
In the end, logic as it used by some apologists is a game, divorced from reality and which they only ‘win’ because nobody else is actually playing. Logic is far more complex a device than these simplistic examples would indicate.
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Sapphire – you point out that wonderful thing, the casual use of language. It is the bane of anyone who desires to have things be absolute, or for everything to adhere to a strict logical framework (with again, all statements being intended as absolute if they are grammatically formed as being absolute).
Those words “all”, “always”, “never”, “can’t”, etc are real buggers. Thankfully most people operate in the real world and not in those absolute worlds. It’s when the exceptions come up in the philosophical/theological arguments that things start getting bizarre.
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