I wonder whether accounting for overdispersion in a GLM (Quasi - Poisson instead of Poisson family) has an effect on the model checking plots (plot of residuals against fitted values, a scale–location plot of residuals against fitted values, a normal QQ plot and a plot of residuals against leverages). I tested this and apart from the scale in the plots nothing changes.
Is this hazard or is this always the case?
It's always the case -- quasi Poisson is effectively Poisson with an additional scale factor.
The dispersion parameter corresponds to rescaling a Poisson variable by that parameter.
To be more specific, the scale factor I referred above to is the dispersion parameter.
In a quasi-Poisson GLM, $E(Y)=\mu\,$, $Var(Y)=\phi.\mu$.
You can get that by taking a Poisson($\lambda$) and multiplying by $\phi$, such that $\mu=\lambda.\phi$. e.g. $Z\sim \text{Pois}(\lambda)$ and $Y=\phi.Z$; then $Y$ has the required properties for a quasi-Poisson. (To my recollection it's the only member of the natural exponential family that has the right variance function.)
For example, Clark and Thayer [1] say
The Over-Dispersed Poisson distribution is a generalization of the Poisson, in which the range is a constant $\phi$ times the positive integers. That is, the variable $Y$ can take on values ${0,1\phi, 2\phi, 3\phi, 4\phi,...}$. It has a variance equal to $\phi$ times the mean.
People using this to model overdispersed counts don't usually worry too much about that since they look on it as a quasi-likelihood model; they're not looking on the scaled Poisson as a specific distributional choice.
[1]: Clark D.R. and Thayer, C.A. (2004),
A Primer on the Exponential Family of Distributions,
CAS 2004 Discussion Papers 117-148,
Casualty Actuarial Society. Arlington, Virginia
(pdf at https://www.casact.org/sites/default/files/database/dpp_dpp04_04dpp117.pdf)
