It is very common for people who are putting forward a claim to say something like, "you must consider this with an open mind", or if their claim is rejected they will say something like, "well of course you don't believe it, you're closed-minded".
There are many ways that this 'appeal to open mindedness' manifests itself, so let's have a look at why it is not usually a valid argument:
What is an open-minded person?
An open-minded person is someone who is willing to consider ideas, opinions and arguments purely on their merit. If an idea can be shown to be correct then an open-minded person will alter, or add to, their world-view with this new-found knowledge. If the new idea does not stand up to scrutiny however, it will be rejected.
Having an open-mind does not mean embracing all kinds of weird ideas and uncritically accepting them no matter how bizarre or unsupported by evidence they are. The actual word used to describe a person who will believe in absolutely anything is: credulous.
Why do skeptics often get accused of not having open-minds?
Misuse of the term "open-minded".
It is the misuse of the term open-minded that the credulous hide behind. They use it as a shield against the use of critical thinking and logical analysis of their ideas, claims, and often bizarre beliefs.
The "I'm open-minded" claim is used to counter the lack of evidence, or the quality of evidence supporting a claim.
The appeal to open-mindedness is really an appeal to relinquish one's rational integrity. It does not matter how or why something may appear to be true, that there is no credible evidence to support it, or even that it is demonstrably false; it should simply be accepted with an 'open-mind'. In other words, with non-thinking credulity.
Conclusion.
Open-mindedness is considered a virtue, and true open-mindedness is.
The claim to be open-minded is frequently used by people who wish to sound virtuous, and simultaneously make their opponent sound intolerant, while defending or promoting their ideas and beliefs.
This appeal to be 'open-minded' is a fallacious one. Its message is simply: be credulous.