I'm an invited speaker. Can I send a coauthor in my place?

One question is whether your coauthor will do as good a job of speaking or otherwise contributing to the conference. If you are well known as an excellent speaker and your coauthor is not, or if you are a much better known researcher, then the organizers may be unhappy with the substitution. Another potential issue may be if your name has been used to advertise the conference, on posters or the web. However, if you and your coauthor are more or less on an equal footing, then I doubt the organizers will object.

I'd recommend apologizing for being unable to make it due to reasons beyond your control (without going into detail), and offering your coauthor as a potential substitute, while still giving them the chance to decline and just cancel your talk or replace you with someone else entirely. I'd make it clear that you are suggesting the coauthor as a possibility without having asked the coauthor yet, so that they don't feel trapped by not wanting to insult a substitute you have already lined up.

I'd say it depends on what the invitation was like. If all three of you were addressed, and were asked to have someone be the invited speaker, then it's perfectly reasonable to ask to replace yourself; and impolite but tolerable to announce a replacement. If you were invited personally, then - no, it's not appropriate.

However - and this is even more important than what I said above so I'll increase the font size:

It is infinitely better to say you cannot make it because of fatigue or just any excuse, than to collapse and cancel at the last moment, or show up completely unprepared.

in your specific case, it sounds like you're just going to have to postpone something else, even if that means reduced chances of getting a grant, missing a submission deadline etc.

PS - It's a question of ethics as well, not just of etiquette, which is why I retagged. You don't get off that easy...

Unless this is a keynote address (where the organizers specifically want you because you are a bigshot in your field) it is usually sufficient if any of the listed authors presents the paper. Many conferences have "no show" policies which prevent the inclusion of the paper/abstract into the conference proceedings (and hence prevent publication) when none of the authors shows up.