As other answers have noted, you don't lose any money by not accepting the tickets, but you do lose an opportunity.
Is the opportunity worth taking, given that you'll have to spend money in a tight financial situation to do so? That's for you and your wife to decide. When discussing this, keep in mind that the value of things like vacation trips is entirely subjective and emotional, and thus it's not only possible but almost certain that your wife will value the opportunity presented by these tickets differently than you do.
Or to put this in simpler terms, it's entirely possible that your wife really wants or even needs that vacation with you and that she knows that this is her one and only chance of getting it in the near future, even though the timing is really awkward.
Note that this doesn't mean that she wouldn't be able to live without the vacation if she couldn't get it — it just means that, given that she can get it, she feels that the opportunity is worth taking even in your current difficult financial situation. Maybe you don't currently feel the same way; maybe you even feel that the financial difficulties would make it hard to enjoy the trip at all.
You wife probably doesn't want to take you on a trip where you would be unhappy, but on the other hand, presumably you also wouldn't want to deprive your wife of what might be a once-in-a-lifetime experience or a desperately needed break from constant stress. That's why the discussion part is important, and why you'll need to be open and honest with you wife about how the financial stress you're experiencing makes you feel, while also accepting your wife's feelings about the whole situation as valid, however much their might differ from yours.
All that might sound off-topic for a finance Q&A site, but there's actually a way to cast all this in perfectly valid financial terms. Specifically, what you and your wife have here is a limited-time discount on an joint investment opportunity. The return from this vacation investment will be emotional and personal for each of you, but no less real for that.
Your task, together with your wife, is to evaluate this investment in the context of your current financial and emotional situation and to decide whether taking the opportunity now — while it is discounted by the price of the tickets, but while your access to liquid funds is limited — is a better decision than either taking a similar trip later (when your finances might be in a better shape, but you'll also have to pay more for the flights) or forgoing the opportunity entirely.
To do this, you will have to balance financial and emotional costs and benefits. Fortunately, while you cannot generally convert things like happiness, self-actualization, memories, richness of experience or stress into money, you can (more or less reliably) go the other way and estimate the emotional costs and benefits of financial decisions over time. So the question you need to ask is whether your expected enjoyment of the vacation, should you take it, is emotionally worth the stress from the financial hardship it would cause you.
And of course, as this is a joint investment between you and your wife, you'll need to carry out this evaluation together, taking into account also the emotional value each of you derives from the other's happiness and from time well spent together.