So by resigning now, your last day falls within the trip. I think it makes some difference when you were told about the trip.
If it's been in your diary for ages then it's fairly unprofessional to resign now because of the trip, in a way that puts your last day in the middle of the trip. You should have resigned earlier.
If you were told yesterday that you need to go, and your manager knows that these long trips are a problem for you, then responding promptly by resigning today seems pretty reasonable to me.
Either way, probably the technical situation is that you can't refuse to go since it's part of your job duties (and if it's not part of your job duties then you could just refuse to go without resigning!) They might decide not to send you, though, depending whether they'd prefer you to spend your notice period handing over your existing projects/duties to others. Your notice period means: that's how long you can be called on to continue to perform your duties.
However, if the trip is only just announced and they send you despite your resignation, I think you can reasonably expect to be returned home on your final day of notice, not a few days later as the trip is currently planned. Of course there's an opportunity for your employer to mess you about in that respect, if they're feeling angry.
Ultimately they will argue that the company is relying on you to go, and you will argue that if they needed more than two weeks notice of your departure (because they're sending you on two week trips) then they should have contracted you to longer than two weeks notice. If they really want you on the trip, and you really want not to go, then you will not reach a compromise that both of you feel is reasonable and will have to fall back to legal obligations as a measure of "reasonable".
If you can stomach it, though, then to me the obvious thing to do is to resign but offer to work the additional couple of days to the end of the trip if they want you to. If they decline that offer then you've done the noble thing but not had to pay for it. If they accept that offer then think of it as life's punishment to you for not resigning earlier, or not finding a way to establish after the last trip that you will not go on another. Either way you have a better chance of your employer thinking of you as reasonable and professional. This might never matter, but might some day matter quite a lot for some reason.
If you think it's more-or-less certain that they'll send you despite you resigning, then in some sense there's no point resigning immediately. Find your next job first. Then when you do resign you can explain why or not, as you please.