Let $X$ be a metric space with $D⊆X$ a dense subset.

If there is a covering for $D$, under which conditions on the covering is it possible to guarantee that the covering also covers $X$?

For a simple example, if you have that the diameter of the elements of the covering is globally bounded below by a positive number, it is immediate that such a covering will cover the whole space.
And if the diameter is not necessarily bounded below, but for every sequence of elements of the covering with diameter converging to zero, every limit point in X of the sequence is contained in a particular element of the covering, then the covering also covers $X$.