In The Mountain and The Viper we see Tyrion's champion Oberyn lose his trial by combat, and Tywin immediately sentences Tyrion to death.
What I'm wondering is why he did this, was it simply a way to take revenge against Tyrion or is there a rule in Trial By Combat which means you have to be executed if you lose (as the gods have made their wishes known)? Could Tywin still have sent him to The Wall if he so desired?
I've had a look through the rules, but found no mention of this.
The reason Tyrion was not ordered to take the black was not because of the Trial itself. The purpose of the Trial by Combat is clear: to determine guilt or innocence. It does not appear to demand the death penalty if you are found guilty in and of itself, you can likely ask for the trial by combat in any situation where your honor is threatened. The problem is that Tyrion was found guilty of regicide specifically, a crime that is obviously going to be punishable by death in any monarchy that supports the death penalty.
Even if ordering Tyrion to take the black is an option, Tywin would never choose it because he hates Tyrion, and because he would look like he was putting his family over justice in a moment of weakness.
Finally, the fundamental premise doesn't really make sense. If you want give leniency, you better give it before the combat because after it either the person who you wanted to save is typically dead, or you just lost a valued fighter and no longer need to give leniency anyway (since the accused won). If you want to give leniency it just doesn't make sense to do it AFTER a fight to the death. Not to mention the gods themselves found the person guilty. You would be better off in every case simply giving leniency beforehand.
In the Game of Thrones universe, when somebody requests/demands trial by combat they are taking the decision away from fallible humans and leaving the decision to the infallible Gods (In this case the 7, it is unclear if trial by combat takes places where other religions dominate).
For Tywin to overrule the outcome of the battle would be the equivalent of him declaring himself as above the gods, opening himself to much judgement from the public, regardless of his own beliefs.
By demanding the trial (and Oberyn's subsequent loss) Tyrion forces Tywin's hand leaving him no option but to order his execution.
No where in any of the definitions of Trial by Combat does it mention anything regarding punishment.
A trial is to determine guilt or innocence, trial by combat makes "The Gods" the judges, sentencing is a separate process to decide what the punishment is. Presumably "The Gods" are more than capable of doling out punishment if they see fit.
Tyrion is still going to be sentenced by humans and human law that states the sentence for regicide is death, "The Gods" have nothing to do with that. But human law also includes the caveat that a person can join the Night's Watch to be abolished of all crimes, by promising to serve in protection of the realm and giving up everything they once held.
If someone can take the Black after being found guilty of regicide is the legal question and, in the AGOT lore, I don't believe there is any precedent either way.
Could "The Crown" (Tywin) allow Tyrion to take the black? Yes.
Will "The Crown" allow Tyrion to take the black? Who knows?
