No, John Carpenter's The Thing was not influenced by At the Mountains of Madness in any significant way. It is a very faithful adaptation of the novella Who Goes There? by John W. Campbell, which is now available online at no cost. The most readable version I can find is this one, in PDF format.
The Carpenter film is so loyal to the original material that even the main characters' names are the same, although there are far more characters in the novella than the movie, and the movie character Childs does not appear in the novella.
John Carpenter has actually said that he saw the previous adaptation of Who Goes There?, the classic 1951 film The Thing From Another World, when he was young, and was disappointed by how much it deviated from Campbell's story. When, many years later, Carpenter decided to make his own adaptation of Campbell's novella, he was totally committed to making it as faithful as possible to the original source material, unlike The Thing From Another World, which bore little resemblance to the Campbell story.
If you watch the movie again, you'll notice that there isn't much room to insert influences from other sources - the film is incredibly claustrophobic, with a small number of characters, all of them men, in a very confined space. The story unfolds over the course of a few days, but as a viewer, we might be forgiven for thinking that it actually takes place over the course of a few hours. There is very little down time; something terrible happens every few minutes, and no one has time to stop and think about their situation. Once the action begins, it is a roller coaster ride to the finish line.
In short, introducing elements from other stories, including At the Mountains of Madness, would have been contrary to Carpenter's vision and mission in making the film, and even if he had wanted to introduce such outside influences, he had very little opportunity to do so. He really needed every last second of screen time to tell the story he set out to tell. Regardless of how much he enjoyed At the Mountains of Madness (and he clearly enjoyed it a great deal), he didn't want to muddy the waters by mixing his sources.
I have been doing a little reading, checking out a few articles about The Thing, and I just stumbled across something that would appear to be relevant here. While Carpenter has never suggested that At the Mountains of Madness had any influence on his film, it seems that At the Mountains of Madness was published about 2 years before the John W. Campbell novella Who Goes There?, the story upon which The Thing was based. Thus, it is possible that the similarities you have noticed between The Thing and Mountains of Madness were not inserted by Carpenter, but by Campbell. I don't know much about Campbell, but it is certainly plausible that he was inspired by Lovecraft's story.