The following is the concluding paragraph of Chekhov's story "The Bet" in full.
Next morning the watchmen ran in with pale faces, and told him they had seen the man who lived in the lodge climb out of the window into the garden, go to the gate, and disappear. The banker went at once with the servants to the lodge and made sure of the flight of his prisoner. To avoid arousing unnecessary talk, he took from the table the writing in which the millions were renounced, and when he got home locked it up in the fireproof safe.
There is probably more explication in this story than is absolutely necessary. The answer to the question "Why did the banker lock the letter written by the lawyer in a fireproof safe?" is given by the author himself as "To avoid arousing unnecessary talk, he took from the table the writing in which the millions were renounced, and when he got home locked it up in the fireproof safe." I suppose we have to take the author's word that this was the banker's motive. Chekhov establishes that the banker does not have to worry about being suspected of foul play, because in the very first sentence of the above-quoted paragraph he establishes that not one but two witnesses saw the prisoner voluntarily "climb out of the window into the garden, go to the gate, and disappear." The banker goes to the lodge with these two witnesses and verifies that the prisoner has indeed vacated the premises and therefore forfeited the bet.
Chekhov invented those two witnesses to guarantee that the banker could not be suspected of doing what he actually had intended to do, that is, to murder the prisoner in order to get out of paying him all that money. But locking the papers in a fireproof safe "to avoid arousing unnecessary talk" does not sound plausible. It seems that Chekhov wanted to make the reader feel that the banker was doubly secure. He had the witnesses, the fact that the prisoner had fled, and the handwritten papers to produce if any question should come up.
For example, the lawyer might be found dead. He was in poor physical condition. He had no money and no other shelter. If that had happened, there would be a police inquiry, and the police might not be satisfied with the testimony of two watchmen who worked for the man who was about to owe the lawyer a huge sum of money. Chekhov does not say so, but we suspect that the banker had a selfish motive for saving those papers--and saving them in a fireproof safe at that.
Since the reader is in the banker's point of view throughout the entire story, the reader will naturally sympathize with the banker even though he may not be the most admirable man in the world. The reader will want to feel assured that the matter is finally and definitely resolved, that the banker's terrible problem is solved, and that he will only be troubled by a bad conscience. Once we identify with a character, then whatever happens to him happens to us!
Recall that prior to reading the lawyer's letter, the banker had lost most of his wealth. If he were to pay the lawyer the two million, he would become poor. So, the banker conspired to kill the lawyer before he (the lawyer) could win the bet. However, after reading the letter and that the lawyer would renounce the two million, the banker left the room. The banker despised himself more than ever after reading this:
At no other time, even when he lost heavily on the Stock Exchange, had he felt so great a contempt for himself.
Despite feeling contempt for himself, the banker still realizes that it looks awfully suspicious that the lawyer mysteriously disappears shortly before the end of his incarceration when he would collect his winnings. To avoid arousing suspicion (that the banker might have killed or banished the lawyer), the banker keeps the lawyer's letter in a fireproof box. Therefore, he would always have proof that the lawyer renounced the two million and voluntarily escaped from the lodge. (This would keep the banker from being criticized by others or even perhaps imprisoned, the subject of their original conversation.)