In Age of Ultron, Tony says:
Tony Stark: It's biometrics, right? Like a security code? "Whoever is carrying Thor's fingerprints" is, I think, the literal translation.
Thor: Yes, well that's a very, very interesting theory. I have a simpler one: You're all not worthy.
Granted, Ultron bursts in at this point and wreaks some havoc, but Tony's had plenty of access to the hammer before and will probably have more in the future.
I don't imagine it would take much for someone like Tony to be able to replicate Thor's fingerprints, or even an iris scan, or whatever he needs.
Obviously, in the comics, it's literally a magic enchantment on Mjolnir, so this just wouldn't work, but in the MCU, even the "Magic" in Doctor Strange isn't just magic, it's some kind of energy manipulation via extra-dimensional technology.
Though even if having his fingerprint isn't enough to wield Mjolnir, it strikes me as odd that Tony hasn't even tried.
So, given all that, why hasn't Tony Stark tried to use Thor's fingerprint to lift Thor's hammer?