Short answer: No.
I laughed when I first read this claim, but researching this, with the added help of some friends, was quite a journey.
My first interpretation was that someone had decided that bones contained calcium carbonate, that lemon juice contains citric acid, that acid etches away at calcium carbonate, so clearly drinking lemon juice would "melt" your bones like acid weathers limestone. That is rather silly.
So, I set out expecting to be able to show that there was no such evidence. One way to do that is to find an expert who says there is no evidence.
NDTV Food had a similar thought and asked a couple of nutritionists:
Consultant Nutritionist Dr. Rupali Datta says, "There are no scientific studies of any sort that say lemon juice is bad for health." She went on to explain that due to the presence of vitamin C and anti-inflammatory properties, some believe that it may help improve arthritis, but the belief that lemon juice may harm the bones is, according to her, unfounded.
Bengaluru-based nutritionist Dr. Anju Sood echoed Dr. Datta's sentiments on the subject.
Unfortunately, "nutritionist" is a title that is often adopted by non-experts. There's some evidence that Rupali Datta is a qualified dietitian, which is a more trustworthy title, but I still felt uncomfortable relying on this expert's claim. I decided to keep researching.
I found evidence that a deficiency in Vitamin C affected the repair of bones (in guinea pigs), so to the degree that lemon juice protects against a Vitamin C deficiency, it may help maintain bones.
A friend I was discussing this with forwarded an abstract to a (not peer-reviewed) conference poster that I don't have access to read:
47 postmenopausal women were given the equivalent of the juice of one lemon per day for 2 months, and compared to a placebo group. The study looked at a number of measures of bone metabolism
Conclusion: Drinking lemon juice may boost bone metabolic changes involving both bone resorption and bone formation.
Bone resorption? *face-palm* Of course! The vague phrase "melts bones" almost certainly isn't about acid eating away bones at all! I misunderstood the claim. It is really about bone resorption, the normal process where calcium is converted from bone tissue to blood.
So, this study showed that lemon juice both increased the body absorbing bones and the body laying down new bones. This is a normal part of bone remodelling, so it isn't necessarily a bad thing - it may be because the bones are being reshaped to be stronger.
I still wasn't happy, because this was not peer-reviewed, and I couldn't get past the abstract.
But, now, I knew what I was searching for, I found Effects of Lemon Beverages on Bone
Metabolism and Bone Mineral Density in
Postmenopausal Women: A Double-Blind,
Controlled Intervention Study with
Ca-Supplemented and Unsupplemented Lemon
This 2018 study gave 83 postmenopausal women either no treatment, a beverage containing 30mL of lemon juice (LE), or a beverage containing both the lemon juice and calcium (LECA) for five months.
After five months of intervention, the gain in bone mineral density at the lumbar spine was significantly larger in the LECA and LE groups than in the control group.
Generally, the addition of calcium to the treatment got more effect, but the lemon-juice only treatment did as well or better than the control in several measures of bone density.
It is clear that "long-term" (5 month) use of lemon juice in postmenopausal women doesn't leave the bones weaker, but actually leaves them stronger.
Arguably, "bone melting" (i.e. bone resorption) may increase but this is a natural part of bones being reformed and strengthened.