What are some tests of honor shown by Odysseus?
Well, this is a rather difficult question because honor for a Homeric Greek and honor for a modern American are two different things.
For a Homeric Greek, honor is acquired by achieving glorious deeds in battle and by speaking wise words in council. Odysseus had acquired plenty of honor in battle as someone who fought and survived the Trojan War. Another way that a Homeric Greek could gain honor was by being given gifts by their hosts. The people of Phaeacia, for example, give Odysseus lots of valuable gifts as a token of remembrance for his visit to their land.
As for tests of honor shown by Odysseus, it's not really clear what is meant by this question. Does Odysseus test other people's honor? That does seem to occur at various points in the epic.
In Odyssey 9, for example, Odysseus tests the Cyclops to see if the creature will behave respectably towards strangers (namely, he and his men).
Likewise, once Odysseus returns to his native land, he tests both the members of his household and the suitors to see if they will behave in an honorable fashion. He tests the swineherd Eumaeus to see if he remains loyal to Odysseus. Eumaeus proves himself as honorable by almost literally giving Odysseus the shirt off his back.
Odysseus also tests the suitors to see if anyone honorable exists among them. Unfortunately for the suitors, none of them were found honorable and so Odysseus ends up killing them all.