Why the absence of probability distribution for using Quasi-likelihood? We can compare $2\cdot(\text{difference in log-likelihood})$ to a chi-square distribution but why can't we find or invent a new distribution whose test statistics are some function or form involving the quasilikelihood?
 A: Not sure what you mean.  Quasilikelihood fits don't assume any particular underlying distribution; they simply assume, by analogy with generalized linear models, that using an iteratively reweighted least squares fitting procedure with a reasonable link and mean-variance function could give a good description of data.
Venables and Ripley Modern Applied Statistics in S suggest that it might be OK to use an $F$ test to compare quasi-likelihood models:

(note that they don't explicitly say "this is OK in a quasi-likelihood context"). They go on to describe how this is implemented in R:

In fact, there's even an example of this procedure in example("quasipoisson") (which directs to the ?family page:
 d.AD <- data.frame(treatment = gl(3,3),
                    outcome   = gl(3,1,9),
                    counts    = c(18,17,15, 20,10,20, 25,13,12))
 glm.D93 <- glm(counts ~ outcome + treatment, d.AD, family = poisson())
 ## Quasipoisson: compare with above / example(glm) :
 glm.qD93 <- glm(counts ~ outcome + treatment, d.AD, family = quasipoisson())

 glm.qD93
 anova  (glm.qD93, test = "F")

I have always assumed that R's allowance of this test means that one or more members of R-core, who are generally pretty fussy statistically, thought it was OK ("there's no warning sign telling me not to do this, so I guess it must be OK ...")
I can't find much in the way of early references (I don't see anything about model comparison in a quick browse of McCullagh and Nelder's quasilikelihood chapter [9],nor in a quick glance at Wedderburn's 1974 Biometrika paper), nor do V&R say what they mean by "with some caution". This sort of procedure makes properly educated statisticians nervous because (I think) it's based on much more approximate reasoning and with less formal proof (and maybe stronger assumptions) than classic approaches like the Likelihood Ratio test.
Tjur (1998) discusses the issues and justifies the $F$ test on the basis of analogy to other simulations, but certainly doesn't try to state any rigorous results ...

Nevertheless, also in situations where [normality of the observations] is questionable, common sense suggests that it is better to perform this
  correction for randomness of $\hat \lambda/\lambda$ [i.e., uncertainty in the dispersion estimate] — implicitly making the
  (more or less incorrect) assumption that the distribution of
  $\hat \lambda$ is approximated well enough by a $\chi^2$-distribution with
  $n - p$ degrees of freedom — than not to perform any correction at all — implicitly making the (certainly incorrect)
  assumption that $\lambda$ is known and equal to $\hat \lambda$. This suggestion is supported by simulation studies of the behavior of $T$
  versus normal test statistics in case of strongly non-normal
  responses, which will not be reported here (see Tjur 1995).

I'd love to see suggestions for other references supporting/discussing/exploring this approach.

Tjur, Tue. “Nonlinear Regression, Quasi Likelihood, and Overdispersion in Generalized Linear Models.” The American Statistician 52, no. 3 (1998): 222–227.
