How can I efficiently update a literature review (that got outdated due to being rejected three times)?
I recommend that you find another co-author who is willing and able to spend the time that you cannot. An good candidate may be a supervisee (PhD student, postdoc) who needs to get more familiar with the topic anyway. A fresh view may also help to address possible problems with the paper.
I was taught very early in my doctoral studies that "every article has a home". Not necessarily the home you initially target, but any genuine scholarly article can eventually get published somewhere. Most of the time, when you have trouble publishing something, it is because you are shooting for a publication target with relatively high standards, but the high-standard targets don't accept what you are trying to publish. So, the simple solution is to lower your standards. The not-so-simple question, then, is how low you are willing to lower your expected publication standard. Here are some various thoughts:
First and foremost, I recommend to just publish your old article as soon as possible before it gets older rather than hoping to get it updated. Yes, ideally, that would be the best thing to do scholastically, but practically, when I've been in the situation you've described, I've ended up with work 10 years old and never published.
No matter what you decide to do, I recommend that you immediately publish what you have on SSRN as a working paper. (It takes about one hour if it's your first time.) I try to publish all my work-in-progress in semi-complete state on SSRN. That way, people can easily find it on the Web and benefit from the knowledge while I decide or finalize whatever happens to the article. If eventually nothing happens to the article, at the very least it counts as a non-peer-reviewed publication. However, some of my SSRN working papers have been very well cited, which encouraged me to eventually put in the work to finalize and get them published in regular journals.
I'm not sure why you are only targeting conferences. In information sytems, journals are terminal publications and conferences don't usually count very highly in most hiring, promotion and tenure review committees. (Incidentally, I am an information systems professor.) I know that it is recommended to first publish in a conference before publishing in a journal, but actually, I think what is really useful is to submit to a conference (whether accepted or rejected). The reviewer comments you've already received in your rejection are probably far better than the comments you would get if you presented live, so you have the necessary feedback to improve the article for journal publication. For several reasons, I would target a journal immediately rather than a conference, which leads to my next point.
In information systems (and I suspect the same for many other disciplines), conferences are often not ideal targets for literature reviews. One primary reason is that conferences have low page limits, whereas literature reviews usually need longer-than-average page limits. It is harder to do a thorough review in the space allotted. So, literature reviews often get rejected quickly. (I speak from personal experience, both as a review author and as a rejecting reviewer.) What is more, conferences don't offer "revise and resubmit". Their tight deadlines mean that if the initial submission could be improved on, they would normally be rejected, not for lack of potential, but for lack of time in the conference schedule. All of this to say: just because you got rejected from multiple conferences doesn't necessarily mean that your review article is that bad. You might do better at a good quality journal.
Unfortunately, a high-quality journal would probably not accept an outdated review. However, some lower-quality journals might do so. So, depending on what your institution accepts, submit to lower-quality journals until the article gets accepted. I don't know if they would appreciate the "endorsement", but I've found Inderscience to be a good publisher of this kind of journal: legitimate peer-reviewed journals with respectable integrity and just one or two rounds of feasible revisions, but not too demanding and so high acceptance rates for legitimate scholarly work. You can find a journal from that publisher on almost any field in STEM, business or economics. Most of their journals are not highly ranked, though. (One caveat, though: stay away from new journals: only publish this kind of work in established journals, or else you risk publishing in a journal that won't survive.)
One of the other answers suggested finding a co-author like a doctoral student or post-doc to help you update the review, but this isn't always practical, and they don't always do the work you want. I've tried such things in the past myself, and it never quite worked out as I had hoped. So I've learnt that I need to take full responsibility for the publication of my own articles. Unless you yourself are paying someone just to do that kind of work, delegation of research rarely works. So, I recommend not wasting your time with that hope, unless you have the budget to hire an assistant just to do that, which I suspect from your original post that you don't.