Real life examples of difference between independence and correlation It is well known that independence of random variables implies zero correlation but
zero correlation need not imply independence.
I came across plenty of mathematical examples demonstrating dependence despite zero correlation. Are there any real life examples to support this fact? 
 A: Stock returns are a decent real-life example of what you're asking for. There's very close to zero correlation between today's and yesterday's S&P 500 return. However, there is clear dependence:  squared returns are positively autocorrelated; periods of high volatility are clustered in time.
R code:
library(ggplot2)
library(grid)
library(quantmod)

symbols   <- new.env()
date_from <- as.Date("1960-01-01")
date_to   <- as.Date("2016-02-01")
getSymbols("^GSPC", env=symbols, src="yahoo", from=date_from, to=date_to)  # S&P500

df <- data.frame(close=as.numeric(symbols$GSPC$GSPC.Close),
                 date=index(symbols$GSPC))
df$log_return     <- c(NA, diff(log(df$close)))
df$log_return_lag <- c(NA, head(df$log_return, nrow(df) - 1))

cor(df$log_return,   df$log_return_lag,   use="pairwise.complete.obs")  # 0.02
cor(df$log_return^2, df$log_return_lag^2, use="pairwise.complete.obs")  # 0.14

acf(df$log_return,     na.action=na.pass)  # Basically zero autocorrelation
acf((df$log_return^2), na.action=na.pass)  # Squared returns positively autocorrelated

p <- (ggplot(df, aes(x=date, y=log_return)) +
      geom_point(alpha=0.5) +
      theme_bw() + theme(panel.border=element_blank()))
p
ggsave("log_returns_s&p.png", p, width=10, height=8)

The timeseries of log returns on the S&P 500:

If returns were independent through time (and stationary), it would be very unlikely to see those patterns of clustered volatility, and you wouldn't see autocorrelation in squared log returns.
A: Another example is the relationship between stress and grades on an exam. The relationship is an inverse U shape and the correlation is very low even though causation seems pretty clear. 
