Certainly the chief impact of the "antic disposition" or feigned madness that Hamlet adopts on other characters in the play is to greatly concern them as to what is ailing Hamlet. Note that it is Ophelia's terrified reaction to Hamlet appearing in her chamber that convinces Polonius that Hamlet is mad because of his thwarted love for Ophelia. Note what he concludes:
This is the very ecstasy of love,
Whose violent property fordoes itself,
And leads the will to desperate undertakings,
As oft any passion under Heaven
That does afflict our natures.
Polonius is thus led to believe that Hamlet is "mad" because of the way he had commanded Ophelia to not receive his attentions any more. Ophelia is clearly greatly concerned and shocked by Hamlet's madness. Act II scene 2 indicates how Claudius has enlisted two of Hamlet's old friends, Rosencrantz and Guildenstern, to help him ascertain the cause of Hamlet's madness:
...so by your companies
To draw him on to pleasures, and to gather
So much as from occasions you may glean,
Whether aught to us unknown afflicts him thus
That open'd lies within our remedy.
Claudius and Gertrude therefore are concerned by Hamlet's "transformation" and hope to use his old friends to find out the cause and therefore, as Claudius says, cure him.
It is hard not to relate Hamlet's madness to a kind of detective story. Having introduced this new element into the plot of the play, the other characters desperately try to work out what the reason for this madness is and how they can cure it.