Lively Irish dance causes rebel to lose heart (4)
I believe the answer is:
reel
'lively irish dance' is the definition.
(I've seen this in another clue)
'rebel to lose heart' is the wordplay.
'to lose heart' suggests removing the centre (I've seen 'lost heart' mean this).
'rebel' with its centre taken out is 'REEL'.
'causes' acts as a link.
(Other definitions for reel that I've seen before include "blows making one do this?" , "Fishing rod attachment" , "Lively dance or fishing rod controller" , "spin" , "Scottish/Irish dance" .)