Yes, moral codes encourage people to embrace suffering in themselves or towards others.
Morality can cause more suffering than it relieves if it encourages people to embrace suffering in themselves or towards others. This can be detrimental to the well-being of the person embracing suffering and/or the person suffering is directed towards.
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Morality can lead to more suffering than it relieves because some moral codes allow for the embrace of suffering in oneself as well as in others. It begins with the concept of moral certainty, which is the idea that something is probable enough to be considered, without any reasonable doubt, true. Once applied to morality, this often leads to the idea that this one morality is a universal truth. This idea becomes dangerous when two moralities of this same caliber meet. For when they do, the resulting disagreement can generate suffering on account of being rival interpretations.  Take the Crusades as a historical example. The clash between Christianity and Islam resulted from each side having a perceived claim to the holy land. One can make the case that each side's sense of morality allowed them to press on fighting despite the number of people that would die from those incursions. That because both sides felt entitled to that land, they could keep sending soldiers out to die trying to acquire it, doing whatever they needed to do regardless of the cost. In a way, one might say their morality serves to justify their actions, resulting in needless suffering. Additionally, one can make the case that concepts such as utilitarianism also endorse suffering. Utilitarianism is a moral code whose ultimate goal is to maximize well-being. Sometimes this means that sacrificing an individual for the sake of a group is necessary. In the case of the train trolley redirect scenario, a utilitarian would endorse killing one person for the sake of five other people.  Despite having the best intentions, utilitarianism still causes suffering because it still endorses suffering.
Conversely, not all moral codes embrace suffering but instead work towards easing suffering. For instance, Christianity lends itself towards humanitarian efforts. These efforts can come in many forms, one of which is advocacy. One instance that supports this idea is when the archbishop of York spoke out against cutting overseas aid in the UK, saying that they had both a social and moral obligation to aid in eradicating the unnecessary suffering of other people.  The case shows that in advocating for the maintenance of international aid, the moral code held by the archbishop works to ease suffering as opposed to embracing it.