TL;DR: No, lemon juice has not been shown to work. Other treatments have been shown to be more efficacious and safe, making them better candidates for initial treatment of (non-genital) warts.
Wart treatments have had a lot of research - much of it poor - and a number of meta-analyses have been written to evaluate the information. We can see the medical literature progressing over the past decade in this area.
In 2002, Local treatments for cutaneous warts: systematic review complained that the 50 included trials "provided generally weak evidence".
High quality research on the efficacy of various local treatments for warts is lacking
Evidence, which is generally of a poor quality, shows a beneficial effect of topical salicylic acid and contact immunotherapy with dinitrochlorobenzene
Little evidence exists for the efficacy of cryotherapy and no consistent evidence for the efficacy of all the other treatments reviewed
It did not include citric acid.
In 2005, Cutaneous Warts: An Evidence-Based Approach to Therapy looked at the existing research, but did NOT include citric acid as a potential treatment. It recommended topical salicylic acid and physician-administered cryotherapy, with intralesional immunotherapy for large or recalcitrant (non-genital) warts.
In 2012, Treatment of cutaneous warts: an evidence-based review recommended salicylic acid, silver nitrate and glutaraldehyde first or for mild cases; cryotherapy if that doesn't work, and then if none of the above work, "a variety of alternative therapeutic options (topical, intralesional, systemic, and physical destruction) that are generally off-label (not US FDA approved), and whose use is limited by drawbacks or adverse effects."
I haven't been able to read the entire paper, but they do cite Evaluation of the efficacy of 50% citric acid solution in plane wart treatment. (See @JamesChristopher's answer for more.) Thus it appears they did consider citric acid as one of these alternative topical treatments.
However, they don't hold much hope for this third line of attack:
From pooled evidence-based medicine data, it is possible to conclude that significantly higher remission rates may be expected only with cryotherapy and salicylic acid used in combination.
In 2013, Topical Treatments for Cutaneous Wart: An update, explained:
Little has changed in the field over the past 8 years.
It did not explicitly look at citric acid, but looked at other topical acids.
It recommended cryotherapy and salicylic acid, and complained:
Good quality data for most of the other treatments are still lacking.
but admitted that topical immunotherapy with dinitrochlorobenzene has some evidence to support it.
It was fairly generous about other treatments:
Treatment decisions should be made on a case-by-case basis, according to evidence-based medicine criteria, but also considering patient preference and physician experience. So far, the use of nonconventional approaches in selected cases of relapsing and refractory warts seems justified, although scientific evidence is still low.
As James Christopher explains, even if 50% citric acid was effective, lemon juice holds a much lower concentration. Without proper studies of lemon juice, it is an inappropriate treatment - especially when better treatments have proven to work.