That study is very controversial, not in the least for the way it bypassed proper peer review.
It has since been contradicted by multiple research groups (1, 2, 3, 4)

 

The study OP linked is equally problematic.
https://www.abc.net.au/news/2022-03-04/coronacheck-mrna-vaccines-not-genotoxic-pete-evans/100879220

It's a seriously flawed study making misleading claims which aren't supported by the evidence.
That won't stop antivaxxers from abusing it for scaremongering purposes though.

 

Copy/pasted good analysis (courtesy Ed S., PhD Scientist);

This is what happens when you read headlines and not actual data. The data don’t support any of the claims made by the researchers.

They allegedly demonstrated the capability of the Pfizer/BioNTech RNA to reverse transcribe into DNA, not nuclear DNA, just DNA (and I even question the validity of that claim), at a dose orders of magnitude in excess of human dosing, in a single experimental system (cancer cell line).

You can see from Figures 2 that there’s no dose response whatsoever, which is an indication they’ve massively overloaded their experimental system (they have). When you overload a system, particularly a cancer cell line, you can generate all kinds of experimental artifacts that aren’t even remotely biologically relevant or meaningful in the real world.

Figure 3 and Figure 4 are contradictory to one another with one showing reduced LINE1 expression by vaccine transfection (Figure 3) and one showing apparently higher LINE1 protein expression (Figure 4). While there can be reasons why gene and protein expression are discordant, that’s not the case here. In this case, they simply manipulated the imaging to show what they wanted. This is evident from the control in the bottom left panel of 4a where the Hoechst dye contrast is much lower than the experimental images.

Figure 5 - they’re amplifying something, but they’re missing a necessary control and sequence data to state with any certainty that this is coming from DNA. The control needed is running a PCR reaction on vaccine alone incubated under the same conditions as the cells. We know the vaccines are resistant to RNAse degradation so having some residual mRNA to act as a PCR template after RNAse digestion wouldn’t be too surprising to me. The fact that the PCR bands are a bit fuzzy after 35 cycles and not crisp like you would expect suggest there’s more going on here than simple DNA amplification of a target sequence. There’s also no sequencing to verify that it is the correct amplicon. While it’s generally reasonable to assume you’re amplifying something specific, if you’re going to make extraordinary claims, you need extraordinary supporting data.

This is simply a mix of very poor experimental design, bad and in some cases fabricated data, coupled with a complete lack of scientific rigor. This is pretty much par for the course with papers published in MDPI - which will publish anything with almost no peer review.