For Tank:
Nobody really mentioned it, but shields / barriers are a great way to counter Mei. Reinhardt can be super effective against Mei, especially if you have a Kiriko and a Lucio to give you speed and immunity to freeze. Just block her slow with your barrier and whack her with the hammer when she gets close. Orisa is similarly good, since she can escape Mei’s ultimate easily on her own, and even counter it with a well-timed Terra Surge. Zarya has a high survivability against her but can’t really help your teammates as much when your bubble is down. Sigma can also work, if you time your shields and ultimate correctly, but he severely lacks mobility.
For DPS:
You can outrange Mei with a lot of long-distance heroes like Pharah, Widowmaker, Hanzo, and Ashe. Pharah is especially strong, since you can easily outrange Mei’s Endothermic Blaster. Sombra can completely shut down a Mei by taking away her entire kit of survivability. Close combat characters like Reaper and Junkrat can be devastating against her, especially Junkrat since you can Trap and Mine her while she Cryo-Freezes. Symmetra can also be strong if you place her turrets outside of her range, forcing her to use icicles one at a time, plus you can use her Photon Barrier to help block damage during Blizzard.
For Support:
Lifeweaver can shut down a Mei easily. If one of your teammates gets too close to her, you can Lifegrip them away easily. Your petal can also lift allies out of Blizzard and to safety. His dash can make for an easy getaway if you get flanked, and Tree of Life blocks Blizzards line of sight while providing healing. Kiriko is also devastating against Mei. She can easily counter her Blizzard with cleanse and escape with her teleport, while doing high damage at a safe range. Your ability to teleport in and cleanse can save a teammate from certain death. Mei also lacks the ability to climb walls. Zenyatta is similarly good if he remains at a careful distance and uses Orb of Discord as much as possible. His ultimate counters Blizzard completely. Brigitte can be decent against her as well, blocking both her icicles and slow with her shield, while using Shield Dash to escape. Moira can outrange her and Fade away if in danger. Ana can remain at a safe distance while using her Biotic Grenade to deny Mei’s healing, and a well-placed sleep can stop her from flanking.
If all else fails, just try to maintain as much distance between you and Mei as possible while you chip away at her health.