Usually the General Handbook talks about being "temple worthy" rather than specifically holding a temple recommend. This allows flexibility when members live in distant locations where it may be difficult to meet with leaders, or if a worthy member forgets to renew their temple recommend before it expires.
Temple callings, of course, require a current temple recommend. Prospective missionaries must be "worthy of the companionship of the Spirit" (24.3.2). Members must demonstrate worthiness to receive ordinances such as baptism and ordination to a priesthood office (3.4.2). The handbook talks about worthiness to perform certain priesthood ordinances. Worthy members can be assigned to translate patriarchal blessings (38.2.10.5). Church employees must be worthy of a temple recommend (38.8.6).
Worthiness is mentioned as a consideration (among other considerations) when extending any calling (30.1.1). But I don't think there's a list of callings that require temple worthiness.
In general, I would expect bishoprics, stake presidencies, and ward and stake council members to be temple worthy, but I could imagine situations where an inspired leader might call, for example, a less-active member to serve as a counselor or secretary in the elders quorum or Relief Society.
Samuel Bradshaw • If you desire to serve God, you are called to the work.