This is a challenging question. I assume you mean possible weaknesses in the plot, things that happened which are a little hard to believe. I have the highest regard for this story, but I will try to suggest a few parts that puzzled me a little.
First, it was a little strange that a woman would go to that resort place all by herself, especially so long ago, before the Russian Revolution, when women had so little freedom.
Related to that, it seems a little strange that she would be able to make regular trips from her provincial city to Moscow on such a flimsy excuse--that she was having some sort of medical problems that required a specialist. Her husband must have been a real chump.
Third, it is a little hard to believe that she could allow herself to be picked up so easily. Gurov gets her little dog to bark at him, and the next thing we know they're going to bed in a hotel room.
Fourth, it was also a little hard for me to believe that Gurov would travel hundreds of miles to look her up in her hometown and that he could manage to have that interview with her at the theater without running into her husband and without attracting notice from all the provincial townspeople.
Another point that bothered me a little was Chekhov's descriptions of all the times they spent together and the places they visited. Surely there would have been a lot of gossip about them in that resort where people had little to do except watch each other and gossip. Anna Sergeyevna was expecting her husband to come to Yalta. Surely he would have heard some tales about his wife. (Fortunately for her, he decided not to come, and she returned home alone. Chekhov probably devised the husband's "eye trouble" so that he wouldn't have to deal with the husband at Yalta.)