Will my step-up converter module burn if I connect a power hungry device to it?
No-one can know what will happen to your module in that case.

That's because from "Update 2" in my answer on a previous question about the same "XY-016" module, there is evidence of multiple different MT3608 clone ICs being used on those modules. The behaviour could be different depending on the specific clone that you have.

Either don't risk it (which is my recommendation), or try it yourself in controlled conditions (carefully, with fire extinguisher etc. and help nearby) and find out - but your next module might behave differently, if it has a different clone IC.



After more research, I don't think you can draw 2A in your case anyway. That claim in various adverts, is the very best case! That would apply only when there is a small difference between the input and output voltages, even with a real MT3608 IC. The supplier that you linked says:


  Maximum output current: 2A (recommended for use within 1A)


(my emphasis above)

I don't know whether you want a 24V or 28V output from your 5V input, but that is a very large difference between the input and output voltages. Looking at the MT3608 datasheet, the closest I can see is a graph on page 5, showing 5V input, 12V output, and the max current shown in the graph is only 0.8A.

This is likely due to reaching the current limit of the internal MOSFET switch.

With 5V input and 24V (or 28V) output, I would expect a much lower maximum current  than 0.8A before bad things could happen e.g. possible fire from the module, as shown in the video linked in my answer above.