Is “feel positively about...” grammatically correct?

Solution 1:

Modern Grammar considers 'to feel' as a linking verb here.

A linking verb is a verb which links the subject of a clause and a complement.

'Be', 'seem', 'become' are linking verbs as well.

Oxford Living Dictionary https://en.oxforddictionaries.com/definition/complement

defines COMPLEMENT as

"One or more words, phrases, or clauses governed by a verb (or by a nominalization or a predicative adjective) that complete the meaning of the predicate".

So, the only grammatically correct sentence is 'I feel positive about...'.

As about 'positively', it's an adverb and can be used with action or state verbs.

See some examples (https://context.reverso.net):

• Their interrelation may be explained positively or negatively.

• Emphasis on the health programme had positively impacted other areas.

• There is evidence that investment in energy infrastructure positively influences growth.