Without context, I would interpret it this way:
断れない modifies 相手.
断れない相手
the people [you] can't reject
立場上 probably modifies 断れない, although we can't completely rule out the possibility of it working on だめだ.
立場上断れない相手
the people [you] can't reject because of [your] position
話を持ってきた modifies 人.
話を持ってきた人
the people who brought [you] the case(s)
立場上断れない相手 and 話を持ってきた人 are connected with や, and the result is linked with 顔 with の.
立場上断れない相手や話を持ってきた人の顔
the faces of the people [you] can't reject because of [your] position and the people who brought [you] the case(s)
顔を立てる is an idiomatic expression that means something like "to save face."
立場上断れない相手や話を持ってきた人の顔を立てないとだめ
[you] need to save the faces of the people [you] can't reject because of [your] position and the people who brought [you] the case(s)