The word "thing" in this quotation refers to the "higher love" espoused by Raina, as discussed by dymatsuoka. In a sense, though, it also refers to the roles Sergius finds himself compelled to play, as man, aristocrat, soldier, and lover. Sergius find these expectations fatiguing, as he feels constantly obliged to act out a heroic fantasy rather than admit to sometimes being tired, scared, or just wanting to operate on a normal human level and be himself. Through the play, we gradually see him becoming disillusioned with heroic fantasy and longing for an ordinary life.

Sergius's solution to the fatigue is Louka, who, like Raina's Captain Bluntschli, is a pragmatist. Louka is far more interested in a steady income than grand romance. As Sergius grows in self-knowledge through the play, he begins to understand that his relationship with Raina was indeed grounded in the unreal notions of "higher love" and the social expectations that he marry a girl of his own class. rather than genuine compatibility. By choosing to marry Louka instead, he can relax, no longer having to pretend to be a storybook hero.

This line is spoken by Sergius to Louka, a maid in the Petkoff household, in Act II.  The "thing" that fatigues him is "higher love", and the need to keep up a pretense of it "for any length of time".

Sergius, who has just returned from the war, has been reunited with Raina, with whom he is engaged.  The two have just had a rather silly conversation in which they declare how much they have missed each other and describe their utter worship of each other in very flowery terms.  In that conversation, Raina, in a moment of solemn passion, has proclaimed, "Sergius, I think we two have found the higher love...when I think of you, I feel that I could never do a base deed, or think an ignoble thought", to which Sergius has melodramatically responded, "My lady, and my saint!"  After a few more exclamations of devotion, Raina leaves to get ready to go out with Sergius.

Sergius, who admits to being a scoundrel, feels the need for some "relief" after such an emotionally draining scene.  He seeks this relief by flirting with Louka.