How does lawyer-assisted suicide work? What are the ethical complications? What should the offender's lawyer do if his or her client does not wish to appeal their death sentence? Should that choice be respected? If it is, can this be considered lawyer-assisted suicide? Should a lawyer pursue appeals, despite the wishes or his or her client? Or should the attorney represent the client's wishes, even if those wishes lead to certain death?

According to experts (see the link below), two-thirds of death penalty verdicts that are appealed are set aside because of mistakes by defense lawyers or prosecutors. Therefore, the cost of not appealing a death sentence is great, as these appeals can often result in having a death penalty put aside (in other words, the defendant is no longer subject to the death penalty after the appeal). This argues in favor of lawyers filing appeals for defendants on death row. 
There are several steps in appealing a death penalty verdict. First, the defendant in such a case has the right to appeal to the state's highest court, usually called the state supreme court, and they can also appeal to the federal court. A death penalty appeal often is accompanied by a writ of habeas corpus through which the defendant's counsel can raise constitutional or other issues about the way in which the defendant's case was handled. Finally, defendants can also appeal to the governor of the state for clemency. This process can be lengthy, and people can be on death row for several years. The lengthiness of the process may convince some defendants to refuse to appeal their cases.
If defendants waive their right to an appeal, they should be checked for mental competency. That is, experts should determine that the defendants are capable of making informed decisions. If defendants are mentally ill, they cannot make an informed choice about whether to continue to file appeals, so appeals should be filed for them. In many cases, defendants who choose not to appeal could be mentally ill, which supports the idea that lawyers should continue to file appeals for them. 
https://deathpenaltyinfo.org/stories/lack-of-lawyers-blocking-appeals-in-capital-cases