First, consider the legal standard for such a waiver: you have to be competent to do it.  Competency is defined by Dusky v. U.S.: do you have a rational understanding of the charges/issues; do you have the ability to aid in your own defense (i.e. work with a lawyer.)

It should be obvious that if you were competent to make it through trial, you probably meet both criteria.  Indeed, Justice Powell, in Ford v. Wainwright speculated that competency to stand trial likely presumes competencies for execution.  (But the Court ruled that a formal review is needed in these situations.)

And, many do waive their appeal-- for various reasons.  Possibly some accept their guilt.  Some may have other reasons for dying: prison sucks; they're depressed; etc. 

The trick of this question is that the appeal is not for the defendent, it is for society.  It is to ensure that the death sentence is appropriately used (not excessive, prejudiced, etc.)   So in some states, a defendent does not have the right to waive his appeal; appeals are required by law.

So waiving an appeal is not a right anyone has; this is not to say it can't happen-- it's just not your right to do it, it is an option allowed to you after careful review.  The primary (competency) standard is whether the waiver is "knowing, voluntary and intelligent."  It seems logical to infer that the  "intelligent" can only mean an acceptance of your guilt and the punishment as appropriate; but this also presupposes that you are factually guilty (you can't think your punishment is appropriate if you didn't actually do it).  

Note, therefore, that statutory appeals and the absence of a consitutional right to waiver are consistent with the idea that no one can be competent to commit suicide (and we will leave out the case of terminal illness (which, some argue, is identical to the scenario of execution.))

There's a byproduct of this.  Can you be incompetent to be executed?  If you are incompetent to stand trial, trial gets postponed.  Does your execution?

The typical test for incompetency to be executed is (using Texas as an example; they execute the most people) if the defendant does not understand 1) they are to be executed, imminently; 2) the reason they are being executed.

You can see that nowhere in this test does it ask if you are psychotic, or think you are immortal.  If you say, "they want to execute me on Monday because they think I killed that girl, but really it was aliens that did it, and besides, I can't die because God will protect me-- in fact I wish we would do the execution today because I want to bring about the Rapture" then you are competent.  (However, in this scenario, you wouldn't be able to waive your appeal.)

I emphasize this point-- the absence of any psychiatric tests-- because otherwise we have what happened in Texas in April of 2006: The judge thought Steven Staley was incompetent to be executed-- experts had said he was too mentally ill-- so Staley was sent back to the prison psychiatrists to be forcibly medicated to restore him to competency-- to be executed.

This puts doctors in the bizarre ethical quandary of treating someone to kill them. 

Sell v. US made it explicitly ok to forcibly treat someone to restore them to competency to stand trial, because otherwise (remaining incompetent and postponing their trial) violated their constitutional rights to a speedy trial, etc.  But this ruling, in my opinion, therefore made questionable whether you could forcibly medicate people for any other circumstance except  competency to stand trial. 

But it is pretty obvious to me that  trial competency requires a considerably higher degree of cognitive function: it's more than just knowing something, you also have to be able to rationally manipulate information and assist in your own defense.   Execution competency requires none of this, just a binary, yes/no understanding of two questions.  And therefore, it is a rare case  that the forced administration of antipsychotics would get a person to answer these two questions.

Keep in mind that they made it through a trial-- in my opinion, they met a higher competency standard.  So to say they are incompetent to be executed means they must have become sicker after the trial was over. 

I'm not debating the morality of the death sentence.  I'm pointing out that involving psychiatry at the execution stage virtually guarantees the execution will not take place.  If you somehow decide to say that someone is too mentally ill to be executed-- if you go that far-- then I can forsee almost  no circumstance ever that the person will become competent.  It is likely unethical to force medicate someone to be executed; and even if they were, what doctor is going to be sure-- go on record-- they have improved enough? 

Many are going to disagree with me.  They'll say that Ford is really about (moving towards) prohibiting the execution of the mentally ill (not just the legally insane); in the example above, since the person thinks he is immortal, he would not actually understand the ramifications of execution, so could not be executed.  We can debate this all day; I stick to a strict reading of the test because the alternative puts doctors in the position of deciding who is to be executed. 

I've written about doctors staying clear of social policy issues, and this is a good example.  The decision to execute is society's through the legislature and via the judicial system, not doctors' as the philosophe-proxies of the apathetic bourgeoisie.  Doctors are nearly the worst deciders of justice imaginable (and rightly so, as the practice of medicine inherently precludes its application.)