How to interpret qq plot "not on the line"? What does it mean for my model diagnostics if my qq plot looks like this:

As you can see .... "off the line".
 A: It means, the data has non zero mean. Here is the demo, where there are 3 data sets. The first one is generated from standard normal, the second one (the middle plot) is generated with mean $1.0$ and third one (the right plot) is generated with mean $2.0$
set.seed(0)
par(mfrow=c(1,3))
qqnorm(rnorm(1e3))
abline(0,1,col=2,lwd=2)
qqnorm(rnorm(1e3,mean=1))
abline(0,1,col=2,lwd=2)
qqnorm(rnorm(1e3,mean=2))
abline(0,1,col=2,lwd=2)


A: It depends on how you drew the plot and how you determined the line.  
A qqplot for a normal distribution usually uses the standard normal (mean 0, sd 1) as the reference distribution.  If you use the $x=y$ line as the reference line (one option, others are based more on the actual data)  Then the points will only be expected to fall on (near) the line if the observed data come from a normal with the same mean and standard deviation (or variance).  If the observed data comes from a distribution with a different mean but the same (or nearly the same) standard deviation and the reference line is $x=y$ then the plot will look like what you drew.
To account for this you could instead use a reference distribution that has the same mean and SD as the observed data, or use a different line than $x=y$, such as one drawn through $Q_1$ and $Q_3$, or a regression line through the middle half of the points (or others).
