The Thing has rarely been (as far as I can recall) depicted as bleeding even when he fought incredible powerful or vicious combatants such as Tryco Slatterus
(Former Champion of the Universe) Thundra or the Hulk .
To be fair, it seems a bit odd, the Thing also rarely fights opponents who have the ability to cut him, so that might explain why we've rarely seen him bleeding.
If Ben Grimm's body retains the ability to bleed, (and the Marvel Handbook does not indicate it would not)
it is rarely shown no matter who he is fighting.
The assumption is, he is too difficult to cut deep enough to reach anything resembling a vital organ.
It is also safe to assume, his rocky form, if it still uses blood keeps that part of him deeply embedded within his rocky exterior.
While the Marvel Universe's handbook indicates the Thing still has blood under all of his keratin-like armor, his body's physical structure has been enhanced and become far more durable than it would otherwise.
So even if you could penetrate his external armor, his internal structures are also tougher than they would be normally.
The Thing possesses superhuman strength, endurance and durability.
The mutagentic, specific-frequence cosmic ray bombardment caused his musculature, bone structure, internal organ composition, soft tissue structures and skin to greatly increase in toughness and density.
Despite his bulk, his reflexes have retained their above average human levels.
The Thing's lungs are of greater volume and efficiency than a normal man's enabling him to hold his breath underwater for nine minutes.
He can withstand extremes of temperature from -75 to 800 degrees Fahrenheit for up to an hour before heat prostration occurs.
He can withstand the effects of armor-piercing bazooka shells (15 pounds of high explosives) with no injuries.
REF: Handbook of the Marvel Universe :
Ben Grimm - Thing
I would suspect this is another case of the character's ill defined powers being poorly depicted and almost never explained, thus we are left to wonder how their physiology does what it does.
I equate this along the same lines as Colossus' osmium-transplanted flesh where his body "somehow" (insert handwavium here) trades places with a biological equivalent of osmium (heaviest metal known) and he is able to continue to exist without a problem.
HOWEVER:
In a battle against Wolverine (Fantastic Four vol.
1,
#374 (Mar. 1993)), his face was cut open by Wolverine's Adamantium claws and bled.
The claws left scarring that Ben would later wear a helmet to hide the scars.
He was later healed by Hyperstorm .
scifi.stackexchange.com/users/2765/Thaddeus Howze
Yes
In Fear Itself, while possessed by an Asgardian demon hammer, Thor punched a hole in his chest and while you don't necessarily see blood coming from the wound, Thor's hammer is covered in it.
So that seems to show that he can and does bleed if anyone is actually able to penetrate his skin that far.
scifi.stackexchange.com/users/36023/Matthew Green
Yes, he bleeds.
Here is is bleeding while fighting The Champion.
Notice the trail of blood he leaves as well as the blood coming from his mouth.
The Thing is still "normal" underneath his rocky skin.
He's got blood, organs, everything.
His skin is ~2 inches thick (OHOTMU) and very durable.
That's why it is rare that he bleeds.
scifi.stackexchange.com/users/48696/Race Bannon
No.
In this savage fight against the Hulk, Hulk literally breaks part of his face but he doesn't bleed.
In last week's Future Imperfect 2 Hulk fought another version of the Thing, which less pronounced rocky skin and again broke off chunks of it but Thing didn't bleed.
scifi.stackexchange.com/users/15398/Nullbreaker
In Marvel Two-
In-One #54 (part 2 of six-part Project Pegasus saga), the Thing bled after being shot at point blank range by Deathlok using an amped-up laser:
scifi.stackexchange.com/users/53454/user10034
I'm not sure exactly to what extent his body is composed of  rock .  In order to be alive I'm assuming he has organs and blood and such. His tongue and eyes look human enough and they require blood flow.  But has  Thing 's exterior shell ever been damaged to the point he actually bled?  For the purpose of this question, we're pretending the terrible movies have never existed.
Has Thing ever bled?
scifi.stackexchange.com
<marvel><fantastic-four>
scifi.stackexchange.com/questions/93853
scifi.stackexchange.com/users/20393/Daft
The Thing does bleed. The Marvel Universe's handbook indicates that the Thing has normal blood and organs under all of his keratin-like, rocky armor. For example, Thor punched a hole in his chest and while you don't see blood coming from the wound, Thor's hammer is covered in it. The Thing is made up of blood and organs but his opponents rarely have the ability to make him bleed because of his profound durability. However, Thor succeeded in doing so with a hammer. 
The opponents the Thing fights rarely have the ability to make him bleed. It is hard to make the Thing bleed. One opponent who did cause him to bleed was Thor whose hammer was covered in the Thing’s blood.  The Thing is made up of blood and organs but, despite being "normal" underneath, his physical structure is exceptionally durable.