Retirement-aged cellist who hasn't read a comic in decades (but who enjoys the super hero movie explosion). The question came to me, and my boys are no help... Intense pressure v a tough suit. Thanks.
With the Mark 37 Suit, probably
The Mark 37 suit is:
"Deep Sea Diving Suit designed for extreme pressure and traveling at high-speeds underwater. Features upgraded work-lights for high visibility." ―(Iron Man 3 - JARVIS: A Second Screen Experience) The Mark 37's description in the JARVIS Application.
This is supported by this quote from the Marvel Movies Wikia
Hammerhead [Mark 37] may be made for deep depths and high pressures the other suits may be crushed in.
According to Wikipedia, at the deepest part of the Marina Trench, it exerts
a pressure of 1,086 bars (15,750 psi)
Which is pretty high pressure. We don't actually know, as there aren't the statistics about exactly what pressures this suit can handle, but presumably, as it was designed for the high pressures at the depths of the sea, it may be able to handle it; it certainly has the best chance out of all his other suits!
Given the implied technology and enough time to prepare, i believe he could. we have sent submersibles down there and they would require stronger materials than a man sized object. the biggest problem for him would be sealing the suit, so the water couldn't get in. Again, assuming he had the technology, he has in the comics, yes. With the technology we have in this reality, not likely.
From the comics, yes, beginning with the Mark IV armor. In Iron Man #68, Iron Man swims to an underwater fortress of the Mandarin. A missile from the fortress cracks his eye and mouth shields, forcing him to retreat. Back in the lab, he designs a new mask, reinforced to survive any depth of water. (Although not explicitly stated, presumably the rest of the armor is already strong enough to handle it, otherwise he's upgraded the mask to that strength for nothing.)