Why does the poet feel that death is arrogant in "Death, be not proud"?
Right throughout the poem, the speaker attempts to expose the arrogance and presumptuousness of Death. He does this first of all by personifying Death, thus dragging him down to the level of an ordinary person who can easily be insulted. Once he's done that the speaker is in the ideal position to shower abuse and insults upon Death for his perceived arrogance.
First of all, the speaker thinks that Death is arrogant because he's not as scary or as powerful as many people think. It would seem that the figure of Death has an over-inflated opinion of himself and his capabilities:
though some have called thee / Mighty and dreadful, for thou art not so;
In other words, just because some people have called Death "mighty and dreadful" it doesn't meant to say that he is. The speaker certainly doesn't think so.
The speaker then goes on to belittle Death by comparing him to "rest and sleep". And as there's nothing remotely terrible or frightening about relaxing or dozing off, then there's no need for anyone to be scared of Death.
Furthermore, Death doesn't have as much power as he thinks he has. He is a slave to what must happen, what may happen, to the decisions of kings—because they can send people to their deaths in war or by having them executed—and desperate men, i.e. those who commit suicide. In all these cases it isn't Death that has the power, but fate, chance, and human beings.
In any case, a good Christian like Donne has no need to be afraid as Death is just a short sleep before his soul is rejoined with his body before entering into the bliss of eternity.