Stability of carbocations with -M groups in beta position
These would be the arguments I propose under the assumption that the RDS approximation is valid within the scope of the question:
The order (B &gt; A &gt; C &gt; D)  was said to be correct for the conc. $\ce{H2SO4}$/ $\Delta$ case since there would be no effect of $\ce{CH-}$ acidity since the protonation of the $\ce{-OH}$ group takes place and so the carbocation stability takes precedence meaning that the stronger the -I group, the slower the rate since the transition barrier increases.
However, for the scenario where only heat is used in dehydration (I assume basic conditions here), the reaction proceeds via E1cB  since the two main requirements for E1cB is that there is a relatively acidic hydrogen and a poor leaving group. Here, the RDS of the reaction would be the $\ce{C-H}$ bond cleavage. This would be stabilised by the presence of a strong -I group since a carbanion is being generated here which is stabilised by the delocalisation of the electrons via the inductive effect.
So, for such a scenario the rate order would be D &gt; C &gt; A &gt; B.
The answers depend upon the medium as you can see.
This entire argument seems hand wavy with the assumption taken but otherwise as Martin pointed out, this would not be trivial and would require quantum modelling:

Protonation and deprotonation reactions in a catalytic cycle are notoriously hard to calculate. The RDS approximation will break down.