Long story short: This has already been done by scholars, but...
See e.g. (and for further reading and sources) Gosetti-Ferencei, Jennifer Anna. Heidegger, Hölderlin, and the subject of poetic language: Toward a new poetics of Dasein. Fordham Univ Press, 2004:
It has been pointed out that Heidegger both comes
closer to Hölderlin than any other reader ever has to any poet, and, at
the same time, that "Hölderlin says exactly the opposite of what Heidegger makes him say" (Paul de Alan) - that "Hölderlin's poetry is
turned into its opposite by the use Heidegger makes of it" (Otto Pöggeler). Derrida has suggested that Heidegger's interpretations
are a "catastrophe", and the reader who takes note of the political
undercurrents in Heidegger's interpretations - particularly in the lecture courses on Hölderlin - is puzzled in comparing Heidegger's political gestures to Hölderlin's largely ignored - and in Heidegger's reading almost entirely silenced - political thinking (p. 8)
In essence, the criticism is detailed and spread over a couple of works. It is quite impossible to nail it down in a few paragraphs in its entirety without losing much of the good content produced over the decades on that topic.
One of the more detailed ones with both acceptable length and detail can be found in form of the title-giving essay of Henrich, D., & Förster, E. (1997). The Course of Remembrance and Other Essays on Hölderlin. Stanford University Press :
"The Course of Remembrance” contains perhaps
the most detailed account of Henrich’s multifaceted and complex critique of Heidegger, who himself had devoted a lecture course and an
inﬂuential essay to the interpretation of this poem (p. 10)
The sources I read on the topic agree on that Heidegger is actively making Hölderlin "his" Hölderlin, i.e. uses him for what he himself wants to say instead of even considering what Hölderlin himself has actually thought or meant. He even straight out denied the relevance or need of understanding Hölderlin's philosophical and historical setting, which he justified by his own take on how philosophy (or rather it's destruction) should look like. This encompasses especially the political and theoretical thinking of Hölderlin.
Regarding the theoretical thinking, one must bear in mind that Judgment and Being - Hölderlin's most important fragment on theoretical philosophy, having had great impact in the interpretation of Hölderlin's poetical work as well - had not been published before 1970 and one would be misguided to criticise Heidegger for not taking it into consideration - apart from his methodological reasons for not doing so anyway - though.
This little fragment and its impact could be an additional reason why people think Heidegger got it all the more wrong - from a contemporary perspective. But in general, the interpretating Heidegger did not even want to understand Hölderlin himself, despite the reciting Heidegger showing his magnificent knowledge of Hölderlin's poetry.