What can I do when a TA asks me to marry her so that she can obtain a green card?
I am a U.S. citizen and happen to have a foreign spouse, so I've had some personal experience with this.

It could conceivably work if the two of you had known each other intimately for some time, even if you had a George Bernard Shaw, separate bedrooms kind of relationship.  But if you are just acquaintances, it is unlikely the two of you would get through the interview process successfully.  In separate interviews, you would be asked an unpredictable set of questions about your shared personal history.  There are sample interview questions posted online, e.g. https://www.usavisanow.com/marriage-green-card/marriage-green-card-faq/marriage-green-card-interview/ and http://www.alllaw.com/articles/nolo/us-immigration/what-happens-green-card-marriage-interview.html.

In addition, USCIS (formerly INS) would need to see documentation that your finances have been shared and inextricably intertwined for some time.

You could enjoy the fantasy together for an evening, with the Dépardieu film Green Card.



P.S. I have heard about a businessman's visa, which her budget might be enough for, but I don't know much about it.