Dancing in your example may be interpreted as either a participle or a gerund (or nouny, which I find very pleasing!), but it is most likely to be seen as a verb form because to like here is a catenative verb: it is followed directly by another verb.
As BillJ says, if dancing were premodified (as in occasional dancing), that would force dancing to be seen as a noun form; but without that, dancing by itself, because it directly follows like, is seen as a clause that serves as the complement to like.
While to like can take either the infinitive or the gerund-participle, some other common catenative verbs can take only the marked infinitive, such as:
Others can take only the -ing form:
In this instance, the writer may have chosen the gerund-participle "dancing" in preference to the infinitive "to dance" because there is a slight difference in meaning between the two statements:
I don't like dancing.
This could mean that the speaker dislikes all dancing. Ballet, the waltz, popping and locking, you name it: he doesn't like it.
I don't like to dance.
This means only that the speaker himself dislikes partaking in the activity.