If someone is sentenced to death by a Jewish court and is being held prisoner until the sentence can be carried out, if he escapes, is that a sin for him?
When Beit Din sentences someone to death, although death is a type of atonement, it is still a sin to escape, because Beit Din ruled death on that person Source: Devarim 17,11 ועל המשפט אשר יאמרו לך תעשה - The Judgment which They (the Beis din) tell you shall you do (Just like if one steals and Beit Din rules they must pay back, it is a sin to run away and not pay.) Furthermore, if it says in the Torah that this person must die, (e.g A desecrator of the sabbath must be put to death as it says in Shemos 21,14: ושמרתם את השבת כי קדש הוא לכם מחלליה מות יומת) then a person who escapes would be ignoring a commandment written in the Torah.
Likely not. The responsibility to punish seems to be communal - to put the person to death "so that others may hear about it & be afraid" - or an obligation of the court or king. It doesn't seem to be a personal obligation. As 1040 said though, there seems to be aspects of atonement; for example, the condemned is expected to repent before we exact the penalty, and for a few sins we say that death or exile isn't enough to expiate the sin. Perhaps the story of Achan (Joshua 7) is relevant as well. Perhaps the fact that all Jews suffered for his theft indicates that it is a communal requirement to catch & punish thieves, not just Joshua's or the Elders'. Achan is asked to make a public confession, but the main point of that seems to be to show that [the loss in battle was just &] the lottery method was accurate (Rashi v19).
A sin is not a binary condition. If someone escapes the death penalty with the intent and RESULT that they live a meritorious life, then it is not a sin. Of course if they were guilty that sin still stands.
On the other hand if they kill someone to do it then that is another sin no matter the effect of future events or their prior innocence. If they injure someone in the escape they must ask forgiveness and receive forgiveness to nullify that sin.
If they are wrongfully convicted and in escaping (without harm) they will be able to save the life of others because of their escape, then it is a mitzvah to escape.
Still many grey areas
I would think that since all Jews are obligated to upkeep the Torah laws, including the defendant, by escaping he is not keeping the Torah's law and it should be a sin.
The Source that retribution will come for his sin that he tried to evade justice from is: סנהדרין דף לז: ושם) אר"ש בן שטח אראה בנחמה אם לא ראיתי אחד שרץ אחר חברו לחורבה ורצתי אחריו וראיתי סייף בידו ודמו מטפטף והרוג מפרפר ואמרתי לו רשע מי הרגו לזה או אני או אתה אבל מה אעשה שאין דמך מסור בידי שהרי אמרה תורה (דברים יז, ו) על פי שנים עדים יומת המת היודע מחשבות יפרע מאותו האיש שהרג את חבירו אמרו לא זז משם עד שנשכו נחש Shimon ben Shetach said "I will see redemption did i not see a man running after his friend into an enclosed ruin with a blooded sword in his hands etc. they did not move and a snake came out and bit and killed the guilty man.
This applies with all deaths as the Gemora continues דין ארבע מיתות לא בטלו מי שנתחייב סקילה או נופל מן הגג או חיה דורסתו ומי שנתחייב שריפה או נופל בדליקה או נחש מכישו ומי שנתחייב הריגה או נמסר למלכות או ליסטים באין עליו ומי שנתחייב חנק או טובע בנהר או מת בסרונכי
the punishment of the four death penalties was not abolished. How so? One who was liable to be executed by stoning either falls from the roof or a beast tramples him. That is similar to stoning, which involves being pushed off an elevated place and then stoned. And one who was liable to be executed by burning either falls into a conflagration or a snake bites him, which creates a burning sensation. And one who was liable to be executed by decapitation is either handed over to the ruling monarchy for execution by sword, or bandits attack and kill him. And one who was liable to be executed by strangulation either drowns in a river, or dies of diphtheria [serunki]. Rather, reverse the order of the previous statement: A lion and thieves are cases of harm at the hand of Heaven, while cold and heat are cases of harm at the hands of man.
And since he won't be able to do teshuva as the death will be without prior warning he will also lose his world to come which is not the case when sanhedrin kill him and he does Vidui before death.
The death penalty is atonement. So if he runs away he does not get the proper atonement. All sins do is distance you from the Lord. So does not getting atonement.