Why is Tommen not involved in the trial?
As Tywin tells Tommen shortly after he becomes king:
"Wisdom is what makes a good king. But what is wisdom? How do you know which choice is wise and which isn't? A wise king knows what he knows and what he doesn't. You're young. A wise young king listens to his counselors and heeds their advice until he comes of age. And the wisest kings continue to listen to them long afterwards..."
Which is basically Tywin taking control of the throne and the kingdom. Cersei is Queen Regent until Tommen comes of age, but even she can have little influence in the judgement of the trial, since she was the one to accuse Tyrion - the chief prosecutor, in a way. As the head of House Lannister, Tywin has some degree of authority over her and thence over the young king. In the important matter of Tyrion's trial, he shunts aside all others who might preside over it in order to take that position for himself.
Tommen is still a child, and has little wisdom in such matters (he hasn't even been groomed for kingship, unlike Joffrey), so in any case he would be unfit to take charge of the trial. All he can do is to sit somewhere, quietly observing and learning for the future when he himself might have to preside over such events as king.
Why doesn't he even sit and watch?
The simplest answer is because it's boring. While not as petulant and obnoxious as his older brother, Tommen is nevertheless a prince, brought up in luxury by Cersei Lannister, so he's probably another spoiled brat.
Not only that, but he hasn't even been preparing for kingship all his life, as Joffrey had. He's not used to the responsibility of having to learn kingly behaviour and how to take command - nor has he seen a very good example of this in his 'father' King Robert. Why should he have to sit on his arse in a courtroom and listen to a lot of boring legal proceedings when he could be out doing something interesting? That's just the sort of thing Robert would say, and Tommen could easily be following his example.
Here's a book quote to back this up:
The king sat at the head of the table, a stack of cushions under his arse, signing each document as it was presented to him.
"Only a few more, Your Grace," Ser Kevan Lannister assured him. "This is a bill of attainder against Lord Edmure Tully, stripping him of Riverrun and all its lands and incomes, for rebelling against his lawful king. This is a similar attainder, against his uncle Ser Brynden Tully, the Blackfish." Tommen signed them one after the other, dipping the quill carefully and writing his name in a broad childish hand.
Though he had Joffrey's golden curls and green eyes, the new king shared little else with his late brother. He inclined to plumpness, his face was pink and round, and he even liked to read. He is still shy of nine, this son of mine. The boy is not the man. It would be seven years before Tommen was ruling in his own right. Until then the realm would remain firmly in the hands of his lord grandfather. "Sire," [Jaime] asked, "do I have your leave to go?"
"As you like, Ser Uncle." Tommen looked back to Ser Kevan. "Can I seal them now, Great-Uncle?" Pressing his royal seal into the hot wax was his favorite part of being king, so far.
-- * A Storm of Swords*, Chapter 72 (Jaime)
That doesn't look like a mature and responsible king who takes keen interest in the running of his realm and wishes to learn as much as possible about it.
OK, that covers why Tommen wasn't interested
in watching the trial proceedings. You might still argue that Tywin could have made
him do so. Tywin is evidently trying to educate him to be a wiser king than his brother was or could have been, and part of that should be to encourage him to observe important events that as king he would preside over, so that he can learn for the future.
But there's a lot more to this trial than meets the eye. The various members of House Lannister are up to all sorts of machinations around it. Cersei sees it as a way to get rid of her hated valonqar forever; Tywin's motives are less clear, but it's likely that he also wants to see his younger son convicted (and probably sent to the Wall) so that he doesn't have to proclaim him as his heir to Casterly Rock.
With all that politics in the air, which Tommen is blissfully unaware of, it makes sense to keep him out of the proceedings - he could be a liability. If he was present in the courtroom, he might easily say the wrong thing at the wrong time. Anything he said could be taken pretty seriously, so if he can't be trusted to keep his mouth shut or say only what Tywin tells him to, better he be kept out of it altogether.