How to save face when forced to NOT help?

I think you probably did as well as you could in a difficult situation. It's clear there's a larger underlying issue here ("[your boss] and Person A already discussed this time and time again and that Person A should talk to the project manager Person B and the IT Support manager Person C."); you can't fix that.

At this point, the best thing you can do is to extricate yourself from the situation as much as possible - and to ensure there's a paper trail. If you've got some kind of formal ticketing process for support, ensure all communications go through that rather than direct messages etc. If somebody phones you directly, say you're unavailable and ask them to go through the formal process.

It's unclear from your question but if any of this is going through a personal device of yours rather than an offical work device, simply ignore it.

He insisted and (implicitly) threaten to email the COO about our refusal to cooperate.

Don't worry about this. That's a problem for your boss.

Just forward it to your manager to deal with. An important part of a managers job is to act as a buffer between their team and the rest of the World.

In future just refer them to your manager in the first instance. There is no reason for them to be contacting you and setting up meetings direct unless that is clearly part of your role.

If you’re comfortable that your manager can be relied on to have your back, have a conversation with them about how to handle the situation.

Ask them if they can give you clear (preferably written / emailed) guidelines on how to respond to requests from the other team, or even how to handle informal out-of-band requests in general, and then you can lean on those instructions in future.

For example if you agree a process with your manager where all work requests sent to your team (or all work from Team X, or requests regarding Software Y, or requests to your phone or whatever) requires an official ticket or requires per-task approval from your manager then you can simply reply to informal requests asking them to follow your official process as “your hands are tied and you’re not allowed to work on them otherwise”.

You can’t be (reasonably) accused of obstructing anything then - you’re following explicit instructions set by your manager. If you get complaints or abuse for following your process, refer people to your line manager and forward any abusive messages (including the one about “being rude”).

In an ideal world your manager would also speak to the other team’s management (or the problem individual’s manager) informing them of the new process and make it clear the instruction comes from them and it’s not you being obstructive.

And if your team can publish your process as part of a “team charter” on an internal intranet or knowledge base you can refer people directly to that to read them.

Ultimately this is office politics, and as others have said, it’s your line manager’s job to handle it and let you get on with your day-to-day work…

There is no need at all to "save face". You just tell them "I just talked to my boss, and he said I should only do this job for you if he tells me in person that I should do it. And I should NOT do it just because you ask me.

So I can't help you, unless you talk to my boss and change his mind".