What happens to my account if I forget to order a "sell to close" option?

Let's suppose I put a "buy to open" for AIG at a strike price of $53.50. It expired on Friday June 17, 2016. However, I forgot to "sell to close" before it expired. AIG closed on Friday at $53.42, so there is some value in the option. What would happen to my account? Would I have to buy the 100 shares of stock at $53.42?

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January 2018

Based on your question, I am assuming you bought a call option on AIG with a strike price of $53.5. Since, at expiration the price of AIG was less than the strike price, it will expire worthless. If the price of AIG was above the strike price at expiration, you would have had the right to exercise the option ie purchase AIG at the strike price or just pocket the profit.

January 2018