Be cognizant of time limits. To be persuasive, you also need to be concise. Appellate courts force attorneys to be succinct by imposing time limits. Different courts have different policies. In the First Department, time limits are relatively strictly adhered to, with colored lights signifying that you are running out of time. In the Second Department, they are more prone to being flexible with time limits, but you need to get the hint when the judges are suggesting that you sit down.
Don't make unplanned concessions. The panel of appellate judges will ask the questions, and will occasionally, essentially, cross-examine one of the litigators with yes or no questions. Sometimes, a judge is trying to help you strengthen your argument, or is using you to mouth their own argument in your favor. Other times, the judge is trying to make you see the weaknesses of your case, or to concede a fact or point that supports their preferred outcome. I try not to say never, but be extraordinarily careful of conceding any factual or legal issue during your argument unless it is something you have first carefully considered.
I watched an argument last week where one side responded to progressively stronger requests that she sit down with statements such as "one last thing," followed by another minute or two of talking. Her adversary, on the other hand, agreed with nearly every yes or no question that was posed to him, even when the answers didn't help his client and were not things he necessarily had to agree with. The two were actually rather skillful and experienced advocates. One was from a District Attorney's office, the other from a legal aid organization. They both likely knew that they were bending the "rules," but were engaged in passionate and sincere argument (which generally trumps any black-letter rules, so long as it is done respectfully).
Respectfully disagree with a hostile judge. Dealing with a judge who clearly opposes your case is possibly the most difficult part of arguing an appeal. Most often, unless there is also a judge who seems to be on your side and they just decided to play good cop/bad cop, this signals that at least that judge (if not the entire panel) will side against you. At that stage, you can still make sure the judges understand and respect your reasoning, potentially change their minds, and address concerns that might limit or guide their decision so as to avoid setting a bad precedent. Possibly, if your arguments are strong enough, you'll get a dissent. There are judges -- albeit relatively rarely -- who are more quiet, and won't speak up against a colleague during oral argument even if they agree with you. The merit of an appeal is often decided largely based on the briefs, before the parties ever have an opportunity for argument, but there is always a possibility that a strong argument will sway one or more judges in a different direction (or even slightly different direction, issuing an easily distinguishable decision).
"With all due respect," is a phrase that very strongly signals that you disagree with someone, and should be avoided. I recently saw it used where a judge kept cutting the lawyer off without finishing his responses, and the judge quickly respond by saying that "the respect is implied." It often does not help you persuade the judge you are speaking to, or any that agree with him or her, and it's only real purpose is to draw the attention of any other judges on the panel who may be leaning (or teetering) in your favor.
Don't Make Jury Arguments; Unless, Of Course, It's A Jury Issue. Generally, appeals involve questions of law, and the judges are already familiar with the record and the issues. It is not a closing statement, and shouldn't be treated like one. Be concise, straight-forward, and do not overtly appeal to sympathy or other emotive factors. Obviously, this rule is very flexible, depending on what issues you are dealing with. When the panel is reviewing a jury verdict, for example, your argument may sound similar to a dense, evidence-focused, closing statement.
I watched a rather entertaining argument a couple months ago where a local Brooklyn business purchased an investment property at a foreclosure sale. The property had been owned by a rich old woman, there had been some confusion with the water bill, and it eventually went up for sale. The business was a good faith purchaser, and had no knowledge that the woman intended to pay the bill. The old woman, however, wanted the building back because it had sentimental value and she wanted it to be part of a bequest in her will towards a complex for a medical school.
The lower court had put a hold on the property until a hearing could be held, and the business appealed. An overly-cocky lawyer represented the business, and started with a story about how he went to his son's class for career day and explained what lawyers do. The story had some moral about telling the truth, and the argument was that the other lawyer (not the party) was lying. Accusing the other side of lying really didn't seem to sit well with the court. The story was more of a jury oriented technique (which may or may not have worked better in a closing argument), and in an appellate context seemed to make the panel more sympathetic to the other side. In fact, since the issue below was whether the court below had the equitable discretion to set aside the foreclosure, focusing on credibility may have been very poor move. Not helping the Old Lady's side, of course, was a lawyer's argument that he had filed and served documents (which neither the Court nor opposing counsel received), and his refusal to accept even the possibility that he didn't send them. If someone else would have argued with or for him, they could have more comfortably said that even if the papers hadn't been served, it would have been excusable law office failure. The attorney for the old lade didn't escape some harsh questioning, but it was unclear after the argument how the panel would rule.
Don't Attack Your Adversary, The Lower Court, Judges, Or Court Personnel. Another lesson learned from the prior story is that courts do not like when one lawyer attacks another. If you think the other lawyer deserves to be chastised, let the court do it. This is true in all litigation. Even when arguing that another lawyer's conduct is frivolous, there are ways to do it that avoid ad hominum attacks. When judges see that you respect the process, they, in turn, are more likely to show you the same respect.