This phrase advises a healthy skepticism of the written word.
Is there a similar idiom that advises skepticism of the spoken word?
Gregory Titelman, Random House Dictionary of America' Popular Proverbs and Sayings, second edition(2000) identifies a number of related expressions conveying this idea under the general title "Believe only half of what you see and nothing you hear":
Believe only half of what you see and nothing you hear. Question everything, especially rumors. The proverb has been traced back to Proverbs of Alfred (c. 1300). First attested in the United States in 1770. In 1845, it was used by the American poet Edgar Allan Poe (1809–49). It is found in varying forms: Believe nothing you hear and only half of what you see; Do not (don't) believe anything (everything) you hear nor (and) half of what you read; Never believe anything (everything) you hear; Believe only what they do and nothing of what they say; One must not believe all he hears; You can't believe everything you hear; Don't believe everything you read in the papers, etc. Usually they are followed by another old saying: "But I saw it. I saw it with my own two eyes."
For your purposes, "You can't believe everything you hear" may be the most relevant idiomatic form that offers a direct oral/aural counterpoint to the warning "Paper does not refuse ink" (which is essentially a fancier, more figurative way of saying, "Don't believe everything you read in the papers").
Wolfgang Mieder, A Dictionary of American Proverbs (1992) has some interesting alternative expressions along similar lines:
They say is a tough old liar. [also recorded as "The biggest liar in the world is they say.")
Saying so don't make it so.
You may also like talk is cheap:
: used to mean that it is easy to say that one will do something
(where used here means is used to as opposed to implying an archaic meaning)
There's an idiom that goes not worth the paper it's written on:
: not of real value : not legally valid
(A variation is not worth the paper it's printed on.)
This has a similar connotation to the paper does not refuse ink saying from the question—which leads into my actual answer.
Lee Gesmer, a lawyer, wrote a blog post that added a clever twist to this sentiment in "An Oral Agreement is Only as Good as the Paper It's Written On." Meaning, in short, if it's not put into writing, then it's just words that that don't have any value.
For the purpose of this question, it could be modified slightly:
The spoken word is only as good as the paper it's written on.
The elegance of this is that not only does is it based on an existing idiom in relation to the written word, but it turns it around to refer to the skeptical nature of speech.
The two idioms actually go well together:
Paper does not refuse ink, and the spoken word is only as good as the paper it's written on.
In short, be skeptical of anything, written or spoken.
Actions speak louder than words, which suggests that speech is less credible than actions, is widely used.
Another expression, with usage that extends beyond English, is A closed mouth catches no flies. It exists in many variants and in many languages, and sources attribute it to multiple people, but there seems to be no consensus about where it comes from.
One of this expression's variants, If you keep your mouth shut, the flies can't get in, is, I believe, a better answer to the question, because it implies that flies will just come in on their own. This suggests that someone talking too much would be caught unawares and would appear foolish (and probably be very uncomfortable) because of the fly influx. On the other hand, A closed mouth catches no flies suggests that someone might actually be trying to catch flies by speaking.
A Google search of A closed mouth catches no flies shows a long list of sites that attribute a range of origins to the expression and its variants:
Additionally, Prosper Mérimée ends the second edition of his novella Carmen, which inspired Bizet's opéra-comique, with the expression En close bouche, n'entre point mouche (which corresponds roughly to the English If you keep your mouth shut, the flies can't get in).
The list goes on. Suffice it to say that this is a widely used idiom.
"Get it in writing!" or "I'd get that in writing" are used to indicate that something you are told orally --especially an agreement or contract-- is not as trustworthy as the equivalent in writing.
-- He says he'll give me my money back if the car breaks down in the first year.
-- I'd get that in writing.
In contrast, "Talk is cheap", suggested by user7661803, shows a preference for actions over words and might even be applied to written words. It depends on what you are trying to promote over speech. (There's also "Silence is golden", but that doesn't really convey skepticism as much as dislike.)
“ The words of a gossip are like choice food...” (Proverbs 26:22 HCSB)
Moral lesson: “Think before you swallow a spoken word into your heart.”
Comparing with the original post:
“Paper doesn’t refuse ink.”
This proverb points out a relationship between the elements of the medium that have no bearing on the worth of the written words. Like most proverbs, a discerning mind must take the obvious truth of the statement and project the moral lesson:
“I must evaluate the truth claims of every written statement with a healthy sense of skepticism.”
This proverb in the Old Testament, uses a similar device regarding the spoken word:
“The words of a gossip are like choice food...” (Proverbs 26:22 HCSB)
As the paper readily absorbs the ink of written words, people gladly taste the message of spoken words, but the discerning mind must evaluate the truth claims of every spoken statement with a healthy sense of skepticism.
The Old Testament proverb goes on to point out superior power of spoken words:
“that goes down to one’s inmost being.”
It might be helpful to research the psychology of this observation, but it seems like the passive stimulation of written words allows the critical functions of the brain to operate more readily than the active arousal of spoken words. Regardless, think before you swallow!
Holman Christian Standard Bible, 2009.
Eavesdroppers seldom hear good of themselves.
This has several variants, and it’s easier for the reader to google the expression than to follow links.
The idea, however, is that when people think that they are not being heard, they speak what they feel, which may or may not be true. The air can no more refuse their voices than paper can refuse ink.