No - I don't believe he deserves to die. As a character, he is foolish, annoying, prattling, canniving and sycophantic, but even with all that, I don't believe he deserves to die. Hamlet comments after his death that he learned his lesson for snooping, but he also comments to his mother that his deed (of killing Polonius) is almost as bad as killing the king. I know this comment is geared more at shocking Gertrude, but I think he knows it should not have happened, even if he does feel somewhat vindicated by it.
Polonius, Polonius, Polonius...what to do about Polonius?!? I have to say that my gut reaction, putting aside 21st century morals and beliefs as much as I possibly can, is that he did deserve his death. He made the choices he made - He put his daughter in harm's way (at that point people were saying Hamlet was crazier than a loon, but Polonius offers his daughter to Hamlet, saying, "Oh, I'm sure he's just acting like a mental patient because he's SOOO in love with Ophelia!"); he sent spies out to keep tabs on his son (even though his own behavior was suspect)...ultimately he made very poor choices that led to his demise.
Probably for me, the biggest reason I say that he deserved it is because of his treatment of Ophelia. Claudius was NOT pushing him to make Ophelia the reason for Hamlet's madness. At no point did the king or queen say, "Hey, Polonius, we would like you to see if Hamlet's is insane because of your daughter." Polonius made that choice because (I believe) he wanted to see an alliance between his daughter and the prince. What better way to secure his position at court and his fortune?
Polonius got what he deserved - Hamlet said it perfectly when he called the person behind the arras a rat.
Deserve? No, not deserve, but I don't think the audience is mourning his death either, they feel as Hamlet does, it's tragic, but he should not have been eavesdropping. Polonius may have wanted to "monitor" Hamlet, but it simply was not his place to do so when he's having a private exchange with his mother. He over stepped his bounds and paid the highest price to do so. Shakespeare writes Hamlet so well that we are apt to feel as he does throughout the play because we feel so connected to him, his plight, his mission.
No, he does not deserve to be murdered. He just happens to be in the wrong place at the wrong time (behind curtain number one). His reason is to report back to the king, but Polonius really believes that there is something wrong with Hamlet and that his actions/words need to be monitored. For one thing, Hamlet was supposed to be with Polonius' daughter, Ophelia. As her father, he would certainly need to know what's going on with Hamlet and if he is really mad or not. His tactics leave much to be desired (sneaking around and eavesdropping), but his heart is in the right place. Also, like many other people who are in positions of power, he might have been trying to stay on the good side of Claudius so as not to jeopardize his own position as his right-hand man. By being Claudius' eyes and ears, he can stay in the king's good graces. He does not deserve to die for that!