Yes, Khlestakov takes money from the mayor. A lot of money. The first instance where he takes money from the mayor is in Act II, shortly after the mayor and Khlestakov meet for the first time:
KHLESTAKOV. Well. I've nothing to do with them. [Reflecting.] I don't see, though, why you should talk to me about your scoundrels or officer's widow. An officer's widow is quite a different matter.—But don't you dare to beat me. You can't do it to me—no, sir, you can't. The idea! Look at him! I'll pay, I'll pay the money. Just now I'm out of cash. That's why I stay here—because I haven't a single kopek.
GOVERNOR [aside]. Oh, he's a shrewd one. So that's what he's aiming at? He's raised such a cloud of dust you can't tell what direction he's going. Who can guess what he wants? One doesn't know where to begin. But I will try. Come what may, I'll try—hit or miss. [Aloud.] H'm, if you really are in want of money, I'm ready to serve you. It is my duty to assist strangers in town.
KHLESTAKOV. Lend me some, lend me some. Then I'll settle up immediately with the landlord. I only want two hundred rubles. Even less would do.
GOVERNOR. There's just two hundred rubles. [Giving him the money.] Don't bother to count it.
KHLESTAKOV [taking it]. Very much obliged to you. I'll send it back to you as soon as I get home. I just suddenly found myself without—H'm—I see you are a gentleman. Now it's all different.
GOVERNOR [aside]. Well, thank the Lord, he's taken the money. Now I suppose things will move along smoothly. I slipped four hundred instead of two into his hand.
Khlestakov goes on to take numerous "loans" from the mayor and others in the town. At first, Khlestakov doesn't realize the humorous mix-up that has occurred- that the mayor and all of the townspeople have mistaken him for the Inspector who is meant to visit and evaluate the town. Instead, he believes that he is about to be put in jail for not paying his hotel bill. In this exchange, he promises to return the money to the mayor once he is back home and able to access his own money.
The mayor, under the false assumption that Khlestakov is the Insepctor he is waiting for, gives money to him freely and pays for all of his bills. He even gives him 200 extra rubles in an attempt to bribe him, or, as he puts it to make things "move along smoothly."
Of course once Khlestakov does realize what is going on, he milks it for all that it is worth, making a fool of the mayor and everyone else who loaned him money throughout the course of the play.