This is to add to @ocram's answer because it is too long to post as a comment. I would treat A ~ B + C as your null model so you can assess the statistical significance of a D-level random intercept in a nested model setup. As ocram pointed out, regularity conditions are violated when $H_0: \sigma^2 = 0$, and the likelihood ratio test statistic (LRT) will not necessarily be asymptotically distributed $\chi^2$. The solution was I taught was to bootstrap the LRT (whose bootstrap distribution will likely not be $\chi^2$) parametrically and compute a bootstrap p-value like this:
library(lme4)
my_modelB <- lm(formula = A ~ B + C)
lme_model <- lmer(y ~ B + C + (1|D), data=my_data, method='ML')
lrt.observed <- as.numeric(2*(logLik(lme_model) - logLik(my_modelB)))
nsim <- 999
lrt.sim <- numeric(nsim)
for (i in 1:nsim) {
y <- unlist(simulate(mymodlB))
nullmod <- lm(y ~ B + C)
altmod <- lmer(y ~ B + C + (1|D), data=my_data, method='ML')
lrt.sim[i] <- as.numeric(2*(logLik(altmod) - logLik(nullmod)))
}
mean(lrt.sim > lrt.observed) #pvalue
The proportion of bootstrapped LRTs more extreme that the observed LRT is the p-value.