Censorship was a prevalent issue in Russia when A Hero of Our Time was written. Given this, did the tsar (or czar) government want the novel to be published? Why or why not?

A Hero of our Time was published in 1839, an era in time in which the author was generally believed to be representing himself and his own beliefs in his output. Lermontov was also writing in czarist Russia, a society in which the opinion of the Czar was paramount and literature was tightly controlled by censors, who sought to protect the reputation of the Royal Family and Russia's social structure.
Given this, Lermontov and his works were subject to scrutiny from censors for several reasons.
1. Lermontov himself was unpopular with the Czar for reasons both literary and otherwise. He objected to the hypocrisy and pretentiousness of Russian high society and often flouted its conventions, supposedly insulting the Czar's daughters and engaging in a forbidden duel, which resulted in his exile to the Caucasus. These personal feuds with the Czar meant that his work came under close scrutiny.
2. He was known to associate with members of the Decembrists, a group who opposed the Czar and the structures of Russian society. He also wrote an extremely controversial poem about Alexander Pushkin in which society's rigidity and pretensions were blamed for the writer's ultimate demise. These expressions of unpopular views brought Lermontov under further scrutiny.
3. In A Hero of Our Time, although many of its tropes are simplistic and unremarkable—the disaffected Byronic hero can be found in many other novels of its time—Lermontov's deeper criticisms of Russian society come to light. High society is dissected as ultimately hypocritical, and its morals as being far from dependable. Bela, who is a princess and representative of the highest echelons of society, eventually succumbs sexually to Pechorin. Princess Mary is described as empty and ultimately valueless, with Pechorin's pursuit of her only an expression of how futile he finds his existence in society. Pechorin, like Lermontov, pursues a duel in defiance of the Czar's orders, and ultimately, he is unable to achieve satisfaction that he has a purpose.
In summary, then, this novel is not in and of itself something so outrageous that it would have been censored if anyone had written it. At the time of writing, however, close connections were drawn between authors' behaviors and their output. Lermontov had already expressed criticism of Russian society in other works and had attracted the unfavorable attention of the Czar.
Because he was known to harbor inappropriate views and to behave in a way contrary to the laws of society, his novel attracted closer scrutiny. Its condemnation of upper-class society is not overtly distinct from the lead character's own disaffectedness, but to allow this novel to be published, by this particular notorious author, would have invited readers to draw deeper interpretations from the material than if the author had been unknown.