When I was at school, we were taught that adjectives in a series must be separated by commas.
The modern trend is to omit the commas as long as the meaning is clear.
Look at the sentences --
A beautiful, little, almond-eyed, Italian girl in a dusty, black dress gave us accurate, helpful information.
A beautiful little almond-eyed Italian girl in a dusty black dress gave us accurate helpful information.
Does the presence or absence of commas make any difference to your understanding of the meaning? (Although I think the version with all the commas is a bit easier to read.)
Whilst in Lyndon's example you can remove the commas without seriously impacting the understanding, the same can't be said for your sentence, miriam: "It is profoundly moving tragicomic and heartfelt with a harsh melodramatic impulse at the climax" would be very difficult to read, particularly at the front end of the sentence.
A good rule of thumb (though not absolute) for the use of commas in English is: Put a comma where you would naturally put a pause in speech.
If you applied that to your sentence, you would probably keep all the commas. You just might remove the one after 'harsh', but you would almost certainly keep the one after 'heartfelt'.
If you removed just the one after 'heartfelt', the sentence would read (bizarrely): "It is profoundly moving,[pause] tragicomic and heartfelt with a harsh,[pause]melodramatic impulse at the climax".