Computer controlled winch

Rather than a remote-control winch, I'd say what you need is simply a remote-control power strip (and, ideally, an IP camera to monitor things). Flip on the outlet for the winch, wait (x) seconds or watch the video to know when, and then flip it back off.

You could very easily script this into Nagios, and even include some elementary logic to try to guess whether the door is already open, but unless your server room has a ridiculous delta-T, paging someone to do it by hand is going to be a lot safer.

wouldnt it be easier to lightly spring load the door (to open, not close), and just use an electromagnet to hold it shut normally ? flip the magnet off and job done.

I would be concerned about the safety issues of such a Rube Goldberg setup. I'd recommend a through-the-wall exhaust fan with louvers that close by their own weight and open with the force of the air blown through them. These can be thermostatically controlled.

Also, your winch, my fan and other possible technological solutions will fail for the same reason that the air conditioning failed if the power goes out. Which reminds me, there are louvers that will open without power based on a set temperature. They use a wax-filled "hydraulic" cylinder. The wax expands and opens the louvers when the temperature rises. They are commonly used in greenhouses. If (since!) your systems are on UPSes they will continue to produce heat. At least this passive system would let the heat escape (the louvers would be mounted high on the wall).

Technical gadgets in all their glory...

The usual solution I'd say is to wire a separate room temperature alarm to the burglar or perimeter alarm and give security the authority to open the door if deemed necessary.

Usually they can actually fix the problem on the spot as well, and they'll have a list of contacts to go through, not relying on you personally to be available, and a written set of instructions to follow. All security systems have temperature sensor options and this needs to be tested periodically (what you test is the actual incident response, not just the alarm system itself).

This makes me sleep well at night. A mechanical arm depending on a series of self-setup monitoring software and hardware, not so much. That would be an interesting secondary addition. If security calls you or someone else in the middle of the night and says they got a temperature alarm, it's always nice being able to confirm this through your own monitoring system - but it should always be separate from the actual room sensor imho.

Also, a backup and/or redundant AC is pretty common even in small one-rack server rooms due to their low cost compared to the alternatives, and a third cooling system from the building itself to help slow down the heat build-up if available.

Try to think about this morally and ignore the temptation to create an amazing hack. I know the question is to find a way to make a winch work but there's also a higher issue at stake here.

If your system is over-heating then you've a far greater risk of fire and some sort of automatic door-opening machine is a terrible idea. Failure of your thermal protection (ie, aircon) is a "shut the server down now" issue. If that server cannot go down, then implement some sort of redundancy but the problem lies with your ability to handle down-time not "how to stop the server exploding".

I can't help but laugh about the whole image of a winch pulling a door open. I've been in hundreds of data centers around the world and have never seen a setup describe such as this.

The standard practice is to shut your servers off to protect them but, honestly, it should never come to that if this is engineered correctly.