I put " do not play with THEIR feelings". I am aware that the "ses" is singular but in modern English is common and acceptable to use THEIR in the singular when you are not sure of the gender of the person. Since this sentence is ambiguous about the gender of the person in question i think THEIR should be accepted.
But you cannot make French a gender-neutral language! "Their sentiments" has to be "leurs sentiments." If Duo were to accept "their" for "his" and "her" then think how much more difficult it would be for us to learn how French possessive adjectives, which (in contrast to English ones) agree in both number and gender with what they modify. To be effective, Duo needs to accept answers on the basis of whether they enable English speakers to learn French, not on the basis of whether they enable English speakers to promote a more gender-neutral English.
Well put, evatenn. I give you a lingot for this comment because I'm bored with eglish speakers who patronise the very language they are trying to learn, in almost every discussion. I agree with the morals in the comment above yours, but this is simply not the place to implement them. I do benefit from reading which other answers people suggest, but I think that as learners, we all lose from the "report for not accepting MY answer" approach.
I'm sorry but I disagree. The phrasing of this exercise shows your opening premise is false: "ses sentiments" is a gender-neutral phrase and so a gender-neutral English translation should be allowed. This Duolingo exercise accepts "its" as the possessive which is nonsensical when talking about a person in English. The correct translation is "their".
https://en.oxforddictionaries.com/usage/he-or-she-versus-they Coolman5001 is correct and the Oxford dictionary has his back... good enough for me.