I am in an uncomfortable situation: Last week, shortly prior to Christmas, I submitted my (me as first-author) manuscript. The journal replied on Dec 23rd, saying that the manuscript needs major revisions (and included their suggestions). The time period for the revision is stated as being 7 days, starting with Dec 24. Obviously, as it was Christmas, I have received hardly any answers from my co-authors, including from my supervisor who has the data that needs to be revised. I have already asked the journal for an extension, but they have not replied. What happens if I miss the deadline?
Unless it is for a special issue with a firm deadline (or a conference), probably nothing happens, though there will likely be a delay in publishing so that it can be rescheduled.
It seems a bit unreasonable, but do the best you can and either resubmit it or give the editor a firm-ish date on which you can.
It is unlikely that you will get a rejection for that alone, except in the special issue or conference situation.
One reason for a short deadline might be that someone, say a reviewer, on their end already missed a deadline while the editor wants to fill and issue and so they are squeezing you. Alas.
And, as the comments imply, there may be something fishy with this journal. Assure yourself that it is reputable.
But, even for a reputable publisher the deadlines for completion can be very firm since the paper can't be pushed to a future issue. It might be possible, however, for them to remove it from the "special" and schedule it for a future release. That might be the best you can hope for.
I have had this before (a ridiculously tight deadline) and sent an email to the editor to ask for a new date. He told me that the journal deadlines were only designed to encourage authors and everyone in the journal basically ignored them anyway. You can hope your editor takes the same view!
But to be sure, why not send an email saying something like "I am so sorry I won't be able to meet that deadline because [reasons]. However we can get the paper back to you on [date]. Please let me know if that will be a problem." They won't reply, and you can use their silence as evidence that they accepted the new date if (and they probably won't) they complain later on.
Ultimately the answer to the title question is You don't. If it's not ready, it's not ready. You can't submit a major revision that doesn't address the reviewer's comments. Therefore, you will simply have to miss the deadline and deal with the consequences.
However, don't assume the worst. The journal wants to publish your paper too, or they'd have given a reject decision. For most journals, what I wrote in this answer applies (I used to work in academic publishing so I know this firsthand):
For context, journals usually give these deadlines because they want to keep the author thinking about the manuscript, and are hoping for a timely revision. The last line about your paper possibly being withdrawn is because there are authors who, after a revise decision, decide not to revise their paper but do not inform the journal. If you exceed the deadline your submission gets marked as inactive. When the journal decides to do spring cleaning and remove all the inactive submissions, that’s when your submission might be withdrawn.
A caveat to this is that if your paper is part of a special issue with a fixed publication deadline, then they might have no choice but to publish without your paper. Since in a comment you wrote "it was already the extended deadline when we submitted", you can't really expect them to hold up the special issue even more for your paper.
One more thing. From your description of your paper's review process, it sounds like you are dealing with an MDPI journal. MDPI is an unconventional publisher - their processes are all hyper-accelerated. Their reviews are fast, their decision time is fast, and the time they give for revisions is also fast. As far as I'm aware, this is because they treat the time taken from submission to publication as a key performance metric (comparatively other publishers won't even measure this metric).
If you are dealing with an MDPI journal, then because their processes are atypical & I don't have an internal view of them, I can't tell what will happen. I would nonetheless guess that they will probably accommodate you, because again they want to publish your paper or they'd have given a reject decision. Also, MDPI's special issues are atypical in that they aren't extra issues of a journal (e.g. a journal that publishes 8 issues a year could have a "special issue" and have 9 issues that year). Instead they appear to be collections of papers on a selected topic with an invited guest editor. If this picture is right, then there is no need for a publication deadline. From their website it indeed looks like there is no publication deadline, only a submission deadline. If so, then they should be able to accommodate you.
A final caveat is that I can see them changing the status to "reject & resubmit", which is effectively the same decision as "revise", it just lets them game their time-to-publication metric. I don't say that they will, but I can see it happening (I can also see it not happening because technically it is past the submission deadline).