What part of speech is the that clause after 'I'm sure'?

[1] It's important [that this is done].

[2] I'm sure [that this will be done].

The lexical class of the bracketed elements is 'declarative content clause'.

In [2] The subordinate clauses combines with the adjective it complements to form a larger adjective phrases functioning as predicative complement of "be". By contrast, in [1] (an extraposed construction) the predicative complement is just "important", just as it is in the basic, non-extraposed, version (see below).

I would strongly recommend dropping the term 'noun clause'. The classification of finite subordinate clauses is based on their internal form rather than spurious analogies with the parts of speech.

Note that in the extrapostion constuction in [1], the subject is the dummy pronoun "it". The that clause is an extraposed subject, but that doesn't mean it's a kind of subject -- it's an element in extraposed position, outside the verb phrase corresponding to the subject of the basic version:

[That this is done] is important.