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In Season 8 episode 5 of Game of Thrones, when Varys hears the sound of the Unsullied coming to his chamber, he burns a letter and removes his rings.

Maybe he burns the letter so that they won't find any evidence against him.

Why remove the rings?

Removing valuable item was a sign of him knowing his coming fate. He knows that he will be executed now or we can say burned alive.

From IGN

When Varys heard a door open and the footsteps of soldiers approaching, that was the sign that his time had come to an end, so he decided to burn the letter he was writing to destroy the evidence of his plan and then prepare himself for death by removing his finery.

Removing one’s valuables before imminent demise is a gesture we’ve seen in other stories before. It’s sort of a ritual of accepting one’s death, a way to prepare for what’s to come.

The same article also covers fan theories too, one of the popular ones is the poison ring left for Tyrion or someone else to finish the job.

Same poison theory is covered by gamespot too and go deeper:

Zooming in on Varys's rings and brightening up the image doesn't reveal any conspicuous blue crystals. In fact, the ring we get the best look at seems to have represented Varys's loyalty to the Targaryen queen--it appears to be this ring from jewelry retailer Mey's official partnership with the show. The retailer calls it an "allegiance ring," and their site claims that Varys, Tyrion, Jorah, and Grey Worm have all worn one since Season 7. It was designed to look like a dragon spine. In taking it off, Varys is symbolically shedding the symbol of his loyalty to Daenerys.

But Varys is still loyal to Jon, who he now knows is also a Targaryen. Even though we didn't see the obvious blue poison crystal as Varys removed all his rings, the show seemed to focus on their removal just long enough that it might be important in the finale.

Varys didn't very convincingly burn his letter. He put it in a jar, and then immediately put the lid on it. As we all know, that quells the fire.

Varys knows he cannot defend himself. He knows the accusations are true. This way, he hopes that someone might find what he wrote and realise the truth he died for.

The rings also come into play here. They identify him and his messages. He might have guessed that he would be executed by fire, which would melt the rings for sure. This way, they remain next to his messages.

Another theory goes even further, claiming that the rings contain the key to decrypt his messages. This makes the rings even more valuable. However, earlier in the episode we see him writing in just plain English.

My expectation was that this all was as a "dead man's switch". He's been writing lots and lots of letters describing who Jon really is, and the episode shows that in detail. The episode also shows his "little birds" visiting his room, and rewards are mentioned. Dany and Tyrion executed Varys, but they seem to have no direct influence over his spy network.

So my assumption is that Varys execution would naturally be followed by his "little birds" delivering the letters and disclosing the secret to the realm. The rings would be related to that - as a payment for the delivery, or possibly to assure authenticity of the message by including a notable token of Varys alongside.

Present facts

As it can be seen during the same S08E05, Varys talked with a girl, whom he addressed as Martha:

Martha: "She won’t eat."

Varys: "We’ll try again at supper."

...

Varys: "What have I told you, Martha?"

Martha: "The bigger the risk, the bigger the reward."

Then, he instructed her to return back to the kitchen where she would be expected.

He looks interested in Daenerys to eat. Why?

Past facts

If we remember the Purple Wedding, a necklace was used to introduce the poison inside the feast.

So, it could be possible that the ring in S08E05 were also manipulated to contain any poison. According to this, the previous dialog would have sense: Varys wanted Martha to poison the Queen.

As extra reasoning, we can find the next dialog in a scene between Ned Stark and Pycelle at season 1:

Ned: "Poison is a woman's weapon."

Pycelle: "Yes. Women, cravens... and eunuchs."

This conversation confirmed that Varys were the one behind the poisoning attemp to Daenerys, using the wineseller from S01E07.

Maybe at the last episode, we will see if Varys movement, leaving the ring at the room, will have any consequences.