Cold air would pour through
If air pressure were equal, and everything were perfectly still, then not much. In a normal environment you get perterbations and vortices that will make some of the cold air and warm air pass through the portal.
When warm air passes through the portal, it is less dense than the cold air on the other side, so it would rise back through the portal. This creates a natural barrier where the warm air passing through would push back.
Not so with the cold side. A bit of cold air that gets through the barrier will suddenly be much heavier than the surrounding warm air and start falling. This would create negative pressure at the portal interface that would suck more cold air through. Eventually you'd have a torrent of cold air pouring through.
The cold air pouring through might push some warm air in the other direction. That air would roll over the lip of the other side of the portal and create a wall-like updraft in the cold air.
All that presumes equal pressure on both sides.
Higher cold side pressure
If the cold side has a higher pressure, then you'd just get the instant heavy cold downdraft, like a bomb cyclone. On the warm side, it would increase air pressure, pushing any weather system away from the portal.
On the cold side, it would draw all of the surrounding air into the area, increasing rotational velocity. Oddly, that rotational velocity would reduce the air pressure around the portal, slowing the transfer of cold air. If it were open long enough, a cyclone would inevitably form.
On the warm side, you'd get a snap freeze on the ground. Weather patterns would be highly dependent upon geography, as the cold air would roll downhill. The downdraft would likely create strong winds and pull warm air with it until the two mixed.
Higher warm side pressure
A more interesting case happens if the warm side has more pressure. The immediate response would be a burst of warm air pushing through and stopping as the warm air on the other side countered the flow with its buoyancy. You'd get a roughly ring-shaped updraft around the edges. The fun starts when someone passes through the ring from the cold side.
Passing through from the warm side wouldn't have a meaningful effect. A burst of warm air would trickle back into the warm air bubble. In the other direction, the aircraft/person/projectile would pull a stream of cold air with it. That stream would create suction where it passed through the portal, perpetuating and strengthening the stream until you wound up with half of the portal pouring cold air while the other half poured warm air.
In this case, the rising warm air on the cold side would immediately create clouds and start precipitating downwind of the portal. Everything would be covered in sheets of ice in no time. On the warm side, you'd still get the cold downdraft, but it wouldn't be as much of a weather emergency.
Sounds like a fun disaster film.
(...) Oslo, Anchorage or **Moscow**(...)
In so-be-it Moscow, portal opens you. $\endgroup$