Influence of the Non-Believer:
Harmony in the Home:
Marriage is seen as a partnership with both spouses contributing equally to its upkeep. If spouses come from two different faiths, they are forced to practice separately, which contradicts the expectations of unity in marriage.
Moreover, if the religion has expectations of certain rituals being performed within a household, that responsibility can be complicated immensely by the inclusion of a non-believer. For example, if the husband is expected to perform certain rituals, but the husband doesn't believe, it is highly unlikely the ritual tradition will continue.
Raising of Children:
When multiple groups of people live in close proximity, assimilation to some degree always occurs. That is to say that each group starts borrowing ideas and customs from other groups. When your group is a minority group, there is a high likelihood that eventually many of your traditions will erode away as practices more common to the area are adopted.
The movement of conservative Muslims to the West is one example: the longer they live in the West, the more likely they are to wear Western clothing. Their children are even more likely to consider traditional practices to be archaic and adopt Western ones such as modern clothing, socializing between genders, appreciation of popular music, and non-use of the Arabic language. To continue with traditional practices is a much more active and conscious choice than in the Middle East where it is the common culture.
For some minority groups, assimilation is seen as a threat to the survival of the community. Many Zoroastrians, who may number as few as 200,000 today, forbid marriage to non-Zoroastrians in the fear that the Zoroastrian spouse will start adopting the customs of the spouse in lieu of Zoroastrian customs. The children then might not be raised as Zoroastrians at all.
The process does not work in reverse, bringing people into Zoroastrianism, because many Zoroastrians do not believe in conversion. You are either born Zoroastrian or you aren't. Thus, it is important for Zoroastrian couples to raise practicing Zoroastrian children so that their traditions – not to mention their cultural identity - continue unadulterated.