Does 1NF require that there can be no duplicate rows?
What was called 1NF in the past is considered nowadays part of the definition of the Relational Data Model itself: each attribute must be a single value, neither composed, nor repeated. When we talk about relations we assume implicitly this fact, since structures with non-flat attributes are not considered proper relations.

Note, however, that there exists alternative models, called non-1NF models or NRC which are based on the concept of relations with possibly other relations as values inside tuples (so it is a kind of Nested Relational Model). Such models, presented in the scientific literature but not implemented in commercial systems, are in some sense a sort of “bridge” towards the late object-oriented data model, that allow more complex kind of data (and that have a concept of association which differs from that of foreign key).

A different aspect is that relations are sets, and not multisets (that is they cannot contains duplicate rows). This is a basic principle of the Relational Data Model: we assume that a relation is just a set, stop. For more details, see this answer.

But you should keep in mind the these are mainly theorical aspects, useful when reasoning about relations, about languages to operate on them (for instance the Relational Algebra), and about theories developed on them (for instance the Normalization Theory).

In practice, however, things can be quite different: for instance in Relational Database Management Systems you can have “relations” without keys or unique attributes, so that they can have duplicate rows. Also, you can have composed data types for attributes of records, like arrays (or, worse, people sometimes put multiple values in an attribute maybe as a single string with commas separating the intended values).