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From my answer to the related question: What are the Wonders and the Wonders Made by Man? we see what the Wonders and Wonders Made by Man are/possibly are. From the quotes on the Q/A we can clearly see Tyrion is highly intrigued by them and would like to see them all.

"A scribe, long dead," said Haldon. "He spent his life traveling the world and writing about the lands he visited in two books he called Wonders and Wonders Made by Man."

"An uncle of mine gave them to me when I was just a boy," said Tyrion. "I read them until they fell to pieces."

"The gods made seven wonders, and mortal man made nine," quoted the Halfmaester. "Rather impious of mortal man to do the gods two better, but there you are. The stone roads of Valyria were one of Longstrider's nine. The fifth, I believe."

"The fourth," said Tyrion, who had committed all sixteen of the wonders to memory as a boy. His uncle Gerion liked to set him on the table during feasts and make him recite them. I liked that well enough, didn't I? Standing there amongst the trenchers with every eye upon me, proving what a clever little imp I was. For years afterward, he had cherished a dream that one day he would travel the world and see Longstrider's wonders for himself.

Lord Tywin had put an end to that hope ten days before his dwarf son's sixteenth nameday, when Tyrion asked to tour the Nine Free Cities, as his uncles had done at that same age.

A Dance with Dragons, Tyrion III

From the same passage we can see that he has seen the Valyrian roads:

Come moonrise, they were back in their saddles, trotting eastward under a mantle of stars. The old Valyrian road glimmered ahead of them like a long silver ribbon winding through wood and dale. For a little while Tyrion Lannister felt almost at peace. "Lomas Longstrider told it true. The road's a wonder."

ibid

We also know he has visited the Wall.

The largest structure ever built by the hands of man, Benjen Stark had told Jon on the kingsroad when they had first caught sight of the Wall in the distance. "And beyond a doubt the most useless," Tyrion Lannister had added with a grin, but even the Imp grew silent as they rode closer. You could see it from miles off, a pale blue line across the northern horizon, stretching away to the east and west and vanishing in the far distance, immense and unbroken. This is the end of the world, it seemed to say.

A Game of Thrones, Jon III

However, I'm not sure what other Wonders he has seen/been too. He even actively turned down going to Qarth to see the walls.

"We could sail to Qarth," Penny went on. "The streets are paved with jade there, my brother always said. The city walls are one of the wonders of the world. When we perform in Qarth, gold and silver will rain down on us, you'll see."

"Some of those ships out on the bay are Qartheen," Tyrion reminded her. "Lomas Longstrider saw the walls of Qarth. His books suffice for me. I have gone as far east as I intend to go."

A Dance with Dragons, Tyrion XI

However, that doesn't mean he doesn't want to see the rest of the Wonders still. Do we know what other Wonders/Wonders Made by Man Tyrion has visited?

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From your list in the linked answer:

Confirmed Wonders

  • The Valyrian roads

As you mentioned in the question, he rode one of these with Illyrio Mopatis.

  • The Wall

He pissed off of it, as mentioned in the question.

  • The Titan of Braavos

He hasn't been to Braavos as of A Storm of Swords. It is never mentioned later that he went to Braavos, so it's unlikely that he has seen this one.

Tyrion could hear the commons shouting out King Joffrey's name. In three years that cruel boy will be a man, ruling in his own right . . . and every dwarf with half his wits will be a long way from King's Landing. Oldtown, perhaps. Or even the Free Cities. He had always had a yen to see the Titan of Braavos.
A Storm of Swords, Tyrion VIII, emphasis mine

  • The triple walls of Qarth

Again, as per your question, nope.

  • The three bells of Norvos

No explicit mention.

  • The Long Bridge of Volantis

He has seen the Long Bridge during his captivity with Jorah Mormont:

The gateway to the Long Bridge was a black stone arch carved with sphinxes, manticores, dragons, and creatures stranger still. Beyond the arch stretched the great span that the Valyrians had built at the height of their glory, its fused stone roadway supported by massive piers. The road was just wide enough for two carts to pass abreast, so whenever a wagon headed west passed one going east, both had to slow to a crawl.
A Dance With Dragons, Tyrion VII

  • The Palace With a Thousand Rooms in Sarnath

No explicit mention, but as pointed out in this answer unlikely, since it has been destroyed.

Possible Wonders

  • The Great Pyramid of Ghis

He was captive as a slave at the end of A Dance with Dragons outside of the gates of Meereen. Arguably, the pyramid was visible, so that might count as visiting. That is, if you consider the pyramid one of the Wonders.

  • The festival city of Chroyane

On their trek down the river Rhoyne, they passed what used to be the city of Chroyane:

The fog clung to them, damp and chilly. A sunken temple loomed up out of the greyness as Yandry and Duck leaned upon their poles and paced slowly from prow to stern, pushing. They passed a marble stair that spiraled up from the mud and ended jaggedly in air. Beyond, half-seen, were other shapes: shattered spires, headless statues, trees with roots bigger than their boat.

"This was the most beautiful city on the river, and the richest," said Yandry. "Chroyane, the festival city."
A Dance with Dragons, Tyrion V

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Tyrion has also visited The Long Bridge of Volantis whilst in the city after being captured by Ser Jorah Mormont and taken there.

Volantis straddled one mouth of the Rhoyne where the river kissed the sea, its two halves joined by the Long Bridge. The oldest, richest part of the city was east of the river, but sellswords, barbarians, and other uncouth outlanders were not welcome there, so they must needs cross over to the west.

The gateway to the Long Bridge was a black stone arch carved with sphinxes, manticores, dragons, and creatures stranger still. Beyond the arch stretched the great span that the Valyrians had built at the height of their glory, its fused stone roadway supported by massive piers. The road was just wide enough for two carts to pass abreast, so whenever a wagon headed west passed one going east, both had to slow to a crawl.

A Dance with Dragons, Tyrion VII


It's worth mentioning that he won't be able to visit the Palace With a Thousand Rooms as it was burnt down by Khal Loso.

Bereft of defenders, Sarnath of the Tall Towers fell to Loso the Lame less than a fortnight later. Not even the Palace With a Thousand Rooms was spared when Khal Loso put the city to the torch.

The World of Ice and Fire, Beyond the Free Cities: The Grasslands

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