These lines come in the third stanza, which apparently celebrates the eternal nature of the emotions depicted in the scene painted onto the side of the Grecian urn. The urn is, of course, used as a symbol of eternal art, as it has preserved beautiful scenes from Greek history for a very long time. The third stanza talks of the speaker's enthusiasm or excitement at seeing the emotion conveyed in this scene which has remained the same since it was first created. The lines you have identified, however, present a slightly negative picture of the scene:
All breathing human passion far above,
That leaves a heart high-sorrowful and cloyed,
A burning forehead, and a parching tongue.
These lines seem to describe the consequences of excessive passion by talking about passion that is "far above" and the results of indulging in such a passion, which are feeling heart-sick and having a "burning forehead" and a "parching tongue."