I don't answer the phone when I don't feel good, but some people feel like the have to (answer it).
If the phone rings and I don't feel good, I don't pick up, but some people feel obligated to.
最後の「answer it」は抜いても大丈夫です。
feel obligated to = ～する義務があるように感じる
電話に出る = answer (the phone), pick up (the receiver)
今は携帯・スマホの時代ですが、まだ「pick up」という人がいると思います。
When he doesn't want to answer the phone, he doesn't, but he makes other people do it.
この場合、次のような言い方ができますよ。
ーWhen he doesn't want to answer the phone, he doesn't, but he makes other people do it.
「彼は電話に出たくない時は出ないくせに、他の人には電話に出させる」
to make other people XX で「他の人にXXさせる」と強制させることを言えます。
ご参考まで！
アメリカ合衆国 
カナダ 
